Delio Mugnolo
email: delio.mugnolo@uni-ulm.de

Serge Nicaise
email: serge.nicaise@univ-valenciennes.fr

WELL-POSEDNESS AND SPECTRAL PROPERTIES OF HEAT AND WAVE EQUATIONS WITH NON-LOCAL CONDITIONS

Keywords: 2000 Mathematics Subject Classification. 47D06, 35J20, 34B10 Non-local conditions, Analytic semigroups, Quadratic forms, Weyl asymptotics

We consider the one-dimensional heat and wave equations but -instead of boundary conditionswe impose on the solution certain non-local, integral constraints. An appropriate Hilbert setting leads to an integration-by-parts formula in Sobolev spaces of negative order and eventually allows us to use semigroup theory leading to analytic well-posedness, hence sharpening regularity results previously obtained by other authors. In doing so we introduce a parametrization of such integral conditions that includes known cases but also shows the connection with more usual boundary conditions, like periodic ones. In the self-adjoint case, we even obtain eigenvalue asymptotics of so-called Weyl's type.

Introduction

Several problems in the applied sciences are modeled by partial differential equations on domains that, as such, require the modeler to impose some assumption on the sought-after solution in order to obtain well-posedness in some function space. Typical are, of course, boundary conditions like Dirichlet or Neumann ones. However, in many physical problems certain different constraints are natural: for example, all equations that are derived from conservation laws -like the Cahn-Hilliard equation or the Navier-Stokes equation -admit the conservation of a certain physical quantity (e.g. mass, barycenter, energy, or momentum) and it is natural to wonder whether already these minimal constraints suffice to obtain well-posedness, at least for a special choice of the initial conditions. While this idea seems to be widely applicable, for simplicity we restrict this paper to the heat and the wave equations on an interval. Instead of usual boundary conditions on the solution u, we investigate the role of non-local, integral conditions like (1.1)

1 0 u(t, x)dx = 0, t ≥ 0,
This amounts to imposing that the moment of order 0 (corresponding e.g. to the total mass, in the case of a diffusion equation for which u denotes the relative density of a mixture) vanishes identically. A heat equation complemented by the above condition has been introduced by J.R. Cannon in [START_REF] Cannon | The solution of the heat equation subject to the specification of energy[END_REF], where well-posedness was investigated by methods based on abstract Volterra equations. While Cannon's work has received much attention by numerical analysts, it has gone largely overlooked by the PDE community, with some notable exceptions (cf. [START_REF] Ionkin | Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions[END_REF][START_REF] Yurchuk | Mixed problem with an integral condition for certain parabolic equations[END_REF][START_REF] Shi | Weak solution to an evolution problem with a nonlocal constraint[END_REF] and the references therein to earlier Soviet literature). The fact that (1.1) only eliminates one degree of freedom still forced Cannon and later investigators to impose a local (say, Dirichlet) condition at one of the endpoints. More recently, it has been observed that the local condition at 0 or 1 may be dropped and replaced by another condition on the moment of order 1, like (1.2)

1 0 (1 -x)u(t, x)dx = 0, t ≥ 0.
Wave and heat equations with (generalizations of) conditions (1.1) and (1.2) have been intensively studied by A. Bouziani and L.S. Pul'kina in a long series of papers that seems to begin with [START_REF] Pul'kina | Certain nonlocal problem for a degenerate hyperbolic equation[END_REF] and [START_REF] Bouziani | Problème mixte avec conditions intégrales pour une classe d'équations hyperboliques[END_REF]. In [START_REF] Bouziani | On the solvability of a class of singular parabolic equations with nonlocal boundary condition in nonclassical function spaces[END_REF], a condition on the moment of order 2 is discussed. In fact, over the last 20 years Bouziani, Pul'kina and their coauthors have discussed a manifold of hyperbolic, parabolic and pseudoparabolic equations with such conditions, mostly by numerical methods. Among others, in [START_REF] Bouziani | Initial-boundary value problems for a class of pseudoparabolic equations with integral boundary conditions[END_REF][START_REF] Dai | A moment problem for one-dimensional nonlinear pseudoparabolic equation[END_REF][START_REF] Guezane-Lakoud | Rothe's method for a telegraph equation with integral conditions[END_REF] several weaker well-posedness results for related parabolic, hyperbolic or pseudoparabolic equations have been obtained by different methods. An extensive list of further papers treating these or similar conditions can be found in the introduction of [START_REF] Dehghan | On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation[END_REF]. A few tentative extensions of the above conditions for heat or wave equations on higher dimensional domains have been proposed in the literature, cf. [START_REF] Mesloub | Mixed problem with integral conditions for a certain class of hyperbolic equations[END_REF][START_REF] Pul'kina | Nonlocal problem with a first-kind integral condition for a multidimensional hyperbolic equation[END_REF]. We are not aware of earlier investigations about the possibility of replacing a condition on the moment of order one by a condition on some moment of higher order. In the companion papers [START_REF] Mugnolo | Diffusion processes on an interval under linear moment conditions[END_REF][START_REF] Mugnolo | The heat equation under conditions on the moments in higher dimensions[END_REF] we have further developed our techniques in order to address these issues.

The main goal of this article is to provide an abstract framework -as general as possible -for studying the one-dimensional heat or wave equation with integral conditions by means of semigroup theory. It turns out that, for the spatial operator we are considering, the associated diffusion equation is the gradient flow of a very simple functional -up to lower order terms, it is simply the L 2 -norm -with respect to some H -1 -type inner product. This is not surprising: for example, it is well-known that the gradient flow associated to the L p -norm with respect to the H -1 (0, 1)-inner product is the porous medium equation with Dirichlet boundary conditions. Also the observation that replacing H -1 (0, 1) by H -1 (T ) permits us to realize the diffusion equation with integral conditions as a gradient flow is not entirely new: for example, it is implicitly used in several articles by F. Otto (see e.g. [START_REF] Kohn | Upper bounds on coarsening rates[END_REF]) to discuss some modifications of the Cahn-Hilliard equation. However, by some direct computations performed in Section 3 it turns out that the H -1 (T )-norm, although natural, gives rise to a gradient flow that agrees with the usual diffusion equation only in a suitable quotient space. This surprising fact seems to show that the usual second derivative is not suitable to be endowed with (nonlocal) moment conditions. Indeed, also the associated second order (in time) evolution equation is not the classical wave equation, but rather a non-trivial generalization that coincides with the usual one only for smooth initial data, cf. Theorem 3.8. In the case of the heat equation, instead, this phenomenon can actually be overcome by invoking the smoothing properties of the analytic semigroups yielded by our approach.

The main difficulty associated with our variational approach is that less regular functions suffer a dramatical deterioration of their properties when integrated by parts even against smooth functions. In turn, this yields that the evolution equation associated with the above mentioned gradient flow is, seemingly, an exotic parabolic equation with no evident physical interpretation, see e.g. Theorem 3.1. In order to get rid of these effects, in all the above mentioned papers (see e.g. [START_REF] Cannon | A Galerkin procedure for the diffusion equation subject to the specification of mass[END_REF][START_REF] Bouziani | Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions[END_REF]) the initial data were assumed to have an artificially strong regularity (and the solutions were only shown to satisfy the equation in a weak sense). Our semigroup approach allows us to avoid this problem and to obtain a classical solution even for rough initial data, cf. Theorem 3.8.

Let us finally emphasize one of the advantages of our approach: It is known that as soon as an evolution equation with homogeneous boundary conditions is governed by a C 0 -semigroup, one can extend the functional setting to find a solution to the same evolution equation with inhomogeneous boundary conditions that is still given by a C 0 -semigroup. This classical method is recalled in Remark 5.7 and shows in turn why it essentially suffices to focus on the condition (1.1), whose interpretation as a condition on the mass may appear less physical at a first glance, as it implies the existence of regions with negative density.

The paper is organized as follows. In Section 2 we introduce the basic notations and prove some technical lemmata along with an integration-by-parts-type formula that will prove quite useful, since we are forced to work in Sobolev spaces of negative order so that the usual Gauß-Green formulae do not apply immediately.

We further discuss a class of quasi-accretive extensions of the second derivative with non-local constraints, but due to technical reasons we have to tackle two subcases: the analysis of the corresponding parabolic and hyperbolic problems will be performed in Section 3. By introducing two parameters (a subspace Y of C 2 and a 2 × 2-matrix K, respectively) we are able to treat an infinite class of non-local constraints.

In some particular cases, the spatial operator is even self-adjoint. In Section 4 we are able to describe the spectrum of several self-adjoint extensions of the "minimal" second order derivative, by more or less elementary techniques. We can show among other things that eigenvalue asymptotics of so-called Weyl's type is still valid for such a class of operators.

We end up with some direct extensions of our results to the case of dynamic integral conditions, see section 5, which in turn by known semigroup theoretical methods allow us to treat inhomogeneous integral conditions.

As observed above, it seems that following the original article by Cannon, in the literature the attention has been devoted mostly to problems in which mixed boundary/integral conditions are considered, see e.g. [START_REF] Shi | Weak solution to an evolution problem with a nonlocal constraint[END_REF] as well as [11, §7.5] and references therein. It should be emphasized that this kind of problems is slightly different from ours: it seems impossible to fix our parameters Y, K in such a way that a homogeneous Dirichlet condition at one endpoint can be recovered. Nor we are able to treat generalizations of (1.1) that have enjoyed some popularity in the literature, like time-dependent, mixed boundary/integral constraints of the form u(t, 0) = 0 and

b(t) 0 u(t, x)dx = 0, t ≥ 0,
for some given function b that satisfies a suitable smoothness assumption, as in the original paper by Cannon [START_REF] Cannon | The solution of the heat equation subject to the specification of energy[END_REF], or even

b(t) 0 u(t, x)dx = 1 b(t) u(t, x)dx = 0, t ≥ 0,
as considered in [START_REF] Lesnic | The solution of an inverse heat conduction problem subject to the specification of energies[END_REF].

Conversely, it seems that the methods in the quoted articles cannot be adapted to our general setting. Furthermore, we could not find any previous reference to a classification of self-adjoint integral conditions as the one we perform in Section 3. Most importantly, our operator theoretical approach is based on energy methods and C 0 -semigroups. To the best of our knowledge, it is the first time these kinds of problems are studied in this framework.

Our technique presents quite a few advantages over alternative methods. First of all, we deliver a unified approach that allows us to treat simultaneously a whole class of integral constraints (including those in (1.1)-(1.2)); secondly, it suffices for us to prove well-posedness of the undamped wave equation to obtain automatically, by perturbation methods and the general theory of C 0 -semigroups, well-posedness of, among others, the telegraph equation and the heat equation. We are also able to enlarge the space on which well-posedness is given, in comparison with previous literature: e.g., in [START_REF] Bouziani | Initial-boundary value problems for a class of pseudoparabolic equations with integral boundary conditions[END_REF] the author needs to impose a compatibility condition on the first moment of the initial value, which we can instead drop.

However, the most important by-product of the semigroup approach pursued by us is that valuable information becomes available about the solution operators, in comparison with other techniques: e.g., we obtain analyticity of the semigroup that governs the heat equation. This in turn yields the fact that the solution u is smooth with respect to both the time and space variables and automatically satisfies additional boundary conditions, cf. Corollary 3.10. This should be compared with the much weaker regularity results obtained in [12, §3] or [8, §5] by means of a Galerkin method. Moreover, in this way we are also able to prove that solutions of the heat equation automatically satisfy infinitely many (non-local) boundary conditions along with the integral ones. Also this observation, which is actually a straightforward consequence of our semigroup approach, seems to be new.

Finally, let us stress that form methods are often very efficient because they allow for an easy proof of further properties. In particular, one is often able to show via the so-called Beurling-Deny conditions that the semigroup, which is a priori only generated in a Hilbert space, actually extrapolates to a range of L p -spaces. It seems however out that our functional framework does not allow for an effective application of the the Beurling-Deny method. A rather different approach has been developed by A. Bobrowski and the first author in order to turn our integral conditions into boundary ones: In this way, it is proved in [START_REF] Bobrowski | On moment preserving cosine families and semigroups in C[0, 1][END_REF] that the heat equation is well-posed in all L p (0, 1)-spaces, p ∈ [1, ∞], and even in C[0, 1], if conservation of moments of order 0 and 1 is imposed.

Added in proof: After the manuscript of this article was submitted we have learned from Fritz Gesztesy that some of the operators studied by us fall into the scope of Krein's theory of self-adjoint extensions, cf. [START_REF] Mugnolo | The heat equation under conditions on the moments in higher dimensions[END_REF] for details. In particular, our spectral results in Corollary 4.7.(i) are weaker than those obtained in [1, § 5], cf. [24, § 6] for further details.

The Bouziani space

If we consider (0, 1) as the torus T , then the test function set D(T ) is in fact the set of smooth functions in [0, 1] such that the derivatives at all orders coincide at 0 and 1. In the same manner we will use the Sobolev space H 1 (T ), by which we denote the subspace of those u ∈ H 1 (0, 1) such that u(0) = u(1) (i.e., of those H 1 -functions supported on the torus). We denote by H -1 (T ) its dual. In view of the decomposition

H 1 (0, 1) ∋ u ≡ u -u(0) id -u(1)1 ⊕ u(0) id +u(1)1 ∈ H 1 0 (0, 1) ⊕ C id ⊕ C1
where id is the identical function, one sees that H 1 0 (0, 1) is a closed subspace of codimension 1 of H 1 (T ) as well as a closed subspace of codimension 2 of H 1 (0, 1). In particular, H 1 (T ) = H 1 0 (0, 1) ⊕ C1 and hence H 1 0 (0, 1) is not dense in H 1 (T ). Therefore, using L 2 (0, 1) as a pivot space, it is not true that H -1 (T ) is continuously embedded in H -1 (0, 1). Indeed, the Dirac functional δ 1 lies in H -1 (T ) and is not trivial there, but it agrees with the 0 functional in H -1 (0, 1). Hence, by duality H -1 (T ) can be identified with H -1 (0, 1) ⊕ C with respect to the orthogonal decomposition (for the inner product (2.19) defined below) (2.1)

H -1 (T ) ∋ φ ≡ φ -µ 0 (φ)δ 1 ) ⊕ µ 0 (φ)δ 1 ∈ H -1 (0, 1) ⊕ Cδ 1 .
We thus regard H -1 (0, 1) as a closed subspace of H -1 (T ). Here µ 0 is the linear functional on D ′ (T ) defined by

(2.2) µ 0 (ϕ) := ϕ, 1 , ϕ ∈ D ′ (T ),
which is bounded on H -1 (T ). We will denote throughout by Id the orthogonal projection of H -1 (T ) onto H -1 (0, 1), and by Id m its restriction to

(2.3) H := {w ∈ H -1 (T ) : µ 0 (w) = 0},
which clearly is an isometric isomorphism. Observe that µ 0 (u) is simply the mean value of u, whenever u ∈ L 1 (0, 1), and with an abuse of terminology we will in fact say that φ ∈ D ′ (T ) has mean zero whenever µ 0 (φ) = 0.

Observe that by construction

(2.4) Ker Id = Span δ 1 .
Now for ϕ ∈ D(T ), we can define a primitive of -ϕ

+ 1 0 ϕ(z)dz by Jϕ(x) := 1 x ϕ(y) - 1 0 ϕ(z)dz dy, x ∈ (0, 1),
and therefore for u ∈ D ′ (T ) we define its primitive P u ∈ D ′ (T ) by

(2.5) P u, ϕ := u, Jϕ , ϕ ∈ D(T ). Note that (2.6) P u, 1 = 0, ∀u ∈ D ′ (T ).
This means that P u is the unique primitive of u of zero mean. Now, by definition (P u) ′ , ϕ = -P u, ϕ ′ =u, Jϕ ′ . But since ϕ is periodic,

Jϕ ′ (x) = 1 x ϕ ′ (y)dy = ϕ(1) -ϕ(x) ∀ϕ ∈ D(T ), or rather (P u) ′ , ϕ = u, ϕ -ϕ(1) u, 1 ∀ϕ ∈ D(T ).
Hence the identity

(2.7) (P u) ′ = u -u, 1 δ 1 , ∀u ∈ D ′ (T )
holds. Although (P u) ′ is not necessarily equal to u, the name "primitive of u" for P u is justified by the fact that (P u) ′ = u on the space of distributions u ∈ D ′ (T ) of zero mean and also on D ′ (0, 1). Note finally that (2.8)

P δ 1 = 0 in D ′ (T ), since P δ 1 , φ = δ 1 , Jφ = Jφ(1)
= 0 which is possibly surprising but is in accordance with (2.7).

Lemma 2.1. Let u ∈ L 2 (0, 1). Then P u is given by

(2.9) P u(x) = Iu(x) - 1 0 Iu(z)dz, ∀x ∈ (0, 1),
where

Iu(x) := x 0 u(y)dy, x ∈ (0, 1).
In particular, P u ∈ H 1 (0, 1) and

(P u) ′ = u in D ′ (0, 1). Moreover, P is bounded from H -1 (T ) to L 2 (0, 1).
Proof. First we remark that for u ∈ L 2 (T ) and ϕ ∈ D(T ), we have

P u, ϕ = 1 0 (Iu) ′ (x)Jϕ(x) dx.
Hence by integration by parts, we obtain

P u, ϕ = 1 0 (Iu)(x) ϕ(x) - 1 0 ϕ(z)dz dx,
since the boundary terms vanish due to (Iu)(0) = Jϕ(1) = 0. This shows (2.9) and that P u belongs to L 2 (T ).

In a second step we first easily check that for ϕ ∈ D(T ), Jϕ is in

H 1 (T ) (it is even in H 1 0 (0, 1)) and that Jϕ H 1 0 (0,1) ϕ L 2 .
According to (2.5) for any u ∈ H -1 (T ), we then get

| P u, ϕ | ≤ u H -1 (T ) Jϕ H 1 0 (0,1) u H -1 (T ) ϕ L 2 , ∀ϕ ∈ D(T ).
As D(T ) is dense in L 2 (0, 1), this shows that

P u L 2 u H -1 (T ) , ∀u ∈ H -1 (T ),
as we wanted to prove.

Let us now introduce the spaces

V := f ∈ L 2 (0, 1) : µ 0 (f ) = µ 1 (f ) = 0 and Ṽ := f ∈ L 2 (0, 1) : µ 0 (f ) = 0 ,
where µ 0 is defined in (2.2) and

(2.10)

µ 1 (f ) := 1 0 (1 -x)f (x) dx = 1 0 x 0 f (z)dz dx.
With this notation, (2.9) reads (2.11)

P u(•) = Iu(•) -µ 1 (u), ∀u ∈ L 2 (0, 1),
and moreover µ 0 (Iu) = µ 1 (u), ∀u ∈ L 2 (0, 1).

Note that Ṽ (resp. V ) is the orthogonal complement in L 2 (0, 1) of P 0 (R)) (resp. P 1 (R)). (Here and in the following we denote by P n (R) the space of polynomials of one real variable with complex coefficients and degree ≤ n.)

Corollary 2.2. The linear operator P is bounded from H -1 (T ) to Ṽ , from Ṽ to H 1 (T ), and from V to H 1 0 (0, 1). Proof. The first assertion directly follows from (2.6) and the previous Lemma. The second and third ones are a consequence of the properties

P f (0) = -µ 1 (f ) and P f (1) = µ 0 (f ) -µ 1 (f ),
that are valid for any f ∈ L 2 (0, 1).

Lemma 2.3. We have u H -1 (T ) P u L 2 + | u, 1 |, u ∈ H -1 (T ).
Proof. For all u ∈ H -1 (T ) we have by (2.7)

u = (P u) ′ + u, 1 δ 1 in H -1 (T ).
Therefore, for ψ ∈ D(T ), ψ ′ ∈ D(T ) and has mean zero, and hence Jψ ′ (x) = -ψ(x)+ ψ(1). Accordingly, by (2.5)

P u, ψ ′ = -u, ψ -ψ(1) = -u, ψ + ψ(1) u, 1 , or equivalently u, ψ = -P u, ψ ′ + ψ(1) u, 1 = DP u, ψ + ψ(1) u, 1 . That implies | u, ψ | ≤ | P u, ψ ′ | + |ψ(1)|| u, 1 |. By the Sobolev embedding theorem, we obtain | u, ψ | ( P u L 2 + | u, 1 |) ψ H 1 (T ) , ∀u ∈ H -1 (T ), ψ ∈ D(T ).
This leads to the conclusion by density.

Lemma 2.4. For all f ∈ H 1 (0, 1) and any c ∈ C, we have

P (Id -1 m (f ′′ ) + cδ 1 ) = f ′ -f (1) + f (0) as an equality of L 2 -functions.
Proof. For shortness we write g := Id -1 m (f ′′ ), which g is well defined because f ′′ ∈ H -1 (0, 1). By (2.7) we have

(P g) ′ = g in D ′ (T ).
By the definition of Id m we have f ′′ = g in D ′ (0, 1), and we deduce that (f ′ -P g) ′ = 0 in D ′ (0, 1). Hence there exists a constant a ∈ C such that

f ′ -P g = a in L 2 (0, 1).
The conclusion follows from (2.6) and (2.8).

Remark 2.5. Note that for f ∈ H 2 (0, 1), we have

(2.12) Id -1 m (f ′′ ) = f ′′ + (f ′ (0) -f ′ (1))δ 1 in H -1 (T )
, and therefore P (Id -1 m (f ′′ )) = P (f ′′ ) in L 2 (0, 1). Hence, for functions regular enough the annoying term Id -1 m u ′′ can be safely replaced by u ′′ . Indeed, the same holds for f ∈ H s (0, 1) for all s > 3 2 , since then f ′′ ∈ H s-2 (0, 1) and 1 ∈ H r 0 (0, 1) = H r (T ) for all r < 1 2 . In the same spirit, we have the next equivalence.

Lemma 2.6. Let f ∈ H 1 (0, 1). Then Id -1 m (f ′′ ) belongs to L 2 (0, 1) if and only if f ∈ H 2 (0, 1) and f ′ (0) = f ′ (1). Proof. If f ∈ H 2 (0, 1) is such that f ′ (0) = f ′ (1)
, then the property Id -1 m (f ′′ ) ∈ L 2 (0, 1) directly follows from (2.12). For the converse implication, we notice that

f ′′ = Id -1 m (f ′′ ) in D ′ (0, 1
). But by assumption the right-hand side of this identity belongs to L 2 (0, 1), and therefore f belongs to H 2 (0, 1). By (2.12), we get that (f

′ (0) -f ′ (1))δ 1 has to belong to L 2 (0, 1), that is only possible if f ′ (0) -f ′ (1) = 0.
By Lemma 2.3 the sesquilinear form defined by

(2.13) (f, g) → 1 0 P f (x)P ḡ(x)dx, f, g ∈ H,
is an equivalent inner product on H. For this reason, we will always endow H with the inner product

(•|•) H := (P • |P •) L 2
introduced in (2.13).

For an arbitrary subspace Y of C 2 we define

V Y := {f ∈ L 2 (0, 1) : (µ 0 (f ), µ 1 (f )) ∈ Y }.
Obviously we have the inclusion V ⊂ V Y for any Y and the identities V = V {0} 2 and Ṽ = V {0}×C . For shortness we introduce the linear maps Γ 1 : L 2 (0, 1) → C 2 and Γ 2 : H 2 (0, 1) → C 2 , defined by

(2.14) Γ 1 u := µ 0 (u) µ 1 (u) , Γ 2 u := -µ 0 (u ′′ ) -u(1) u(1) -u(0) . Remark 2.7.
If Y is an arbitrary subspace of C 2 , then the compact mapping

P Y Γ 1 P Y ⊥ Γ 2 : H 2 (0, 1) → C 2
is surjective, as one can see considering polynomials. In particular there exists u ∈ V Y .

Lemma 2.8. The functional µ 1 : L 2 (0, 1) → C can be continuously extended to L 1 (0, 1), but not to H -1 (T ).

Proof. Assume that there exists C > 0 such that

(2.15) |µ 1 (g)| ≤ C g H -1 (T ) ∀g ∈ L 2 (0, 1).
By direct calculations, we see that

1 -x = 1 2 + 1 π ∞ k=1 sin(2kπx) k ,
where the convergence of the series is in L 2 (0, 1). Then for any n ∈ N * and letting

u n := 2π n k=1 sin(2kπ•) ∈ L 2 (0, 1)
we would have

µ 1 (u n ) = n k=1 1 k .
Moreover by the standard characterization of the H -1 (T )-norm, we have

u n H -1 (T ) ∼ n k=1 1 k 2 1 2
.

Hence (2.15) cannot hold since u n H -1 (T ) is uniformly bounded but clearly lim

n→∞ µ 1 (u n ) = +∞.
Corollary 2.9. The following assertions hold.

(1) The space V and Ṽ are dense in H defined in (2.3).

(2) Let Y be a subspace of

C 2 such that Y = {0} 2 and Y = {0} × C. Then V Y is dense in H -1 (T ).
Proof. (1) Clearly, Ṽ is dense in H. Let us pass to the density of V in H. By Lemma 2.8, µ 1 is not continuous on the closed subspace H of H -1 (T ), either. To conclude the proof, it suffices to observe that the null space of the restriction of µ 1 to Ṽ is V , which is then dense in H by [START_REF] Rudin | Functional Analysis, volume 1 of Higher Math[END_REF]Thm. 1.18].

(2) Before proving the second assertion we observe that if Y = {0} 2 and Y = {0} × C, then either Y = C 2 or there exists α ∈ C such that Y is the set of all (z 0 , z 1 ) ∈ C 2 satisfying (2. [START_REF] Desch | Some perturbation results for analytic semigroups[END_REF])

z 1 = αz 0 .
For the case Y = C 2 , V Y = L 2 (0, 1) and the density is immediate. In the second case we can notice that

V Y = V ⊕ Span {g}, for some g ∈ V Y such that µ 0 (g) µ 1 (g) = 1 α ,
whose existence is guaranteed by Remark 2.7. As µ 0 (g) = 1, we see that

H -1 (T ) = H ⊕ Span {g},
and the density of V Y into H -1 (T ) directly follows the density of V into H.

While by Lemma 2.8 µ 1 is not bounded on H -1 (T ), a weaker continuity property does hold.

Lemma 2.10. There exists C > 0 such that

(2.17) |µ 1 (g)| 2 ≤ C g L 2 g H -1 (T ) , ∀g ∈ L 2 (0, 1).
Proof. The following trace inequality is standard (see for instance [9, comment 1.(iii) at p. 233]):

(2.18) |u(1)| 2 ≤ 2 √ 2 u L 2 u H 1 (0,1) , ∀u ∈ H 1 (0, 1). Now if g ∈ L 2 (0, 1)
, then by Lemma 2.1 P g ∈ H 1 (0, 1) and therefore applying (2.18) to P g, we get

|(P g)(1)| 2 ≤ 2 √ 2 P g L 2 P g H 1 (0,1) .
Now, as Lemma 2.1 shows that (P g) ′ = g, we have

P g 2 H 1 (0,1) = P g 2 L 2 + g 2 L 2
. Moreover P g being given by (2.9), by the Cauchy-Schwarz inequality we see that

P g L 2 ≤ Ig L 2 + 1 0 x 0 g(y)dy dx ≤ 2 Ig L 2 ≤ √ 2 g L 2 .
Hence again owing to Lemma 2.1 we obtain

|(P g)(1)| 2 ≤ C g H -1 (T ) g L 2 , ∀g ∈ L 2 (0, 1),
for some C > 0. Since µ 1 (g) = µ 0 (g) -(P g)( 1), and because

|µ 0 (g)| ≤ g H -1 (T ) g L 2 ,
we can conclude that (2.17) holds.

In view of Lemma 2.3 we see that

(f, g) → (P f |P g) L 2 + µ 0 (f )µ 0 (ḡ), (2.19)
defines an inner product in H -1 (T ) whose associated norm is equivalent to the standard norm of H -1 (T ). We will stick to this inner product on H -1 (T ) throughout this article, and in particular we denote

f 2
H -1 (T ) := P f 2 L 2 + µ 0 (f )µ 0 ( f ). We will repeatedly make use of the following integration-by-parts-type formula.

Lemma 2.11. Let u ∈ H 1 (0, 1), c ∈ C and h ∈ L 2 (0, 1). Then

(Id -1 m (u ′′ ) + cδ 1 |h) H -1 (T ) = c + u(1) u(0) -u(1) µ 0 (h) µ 1 (h) C 2 -(u|h) L 2 .
Proof. Set g = Id -1 m (u ′′ ) + cδ 1 , then by Lemma 2.4 we have

P (g) = u ′ -a,
where a := u(1)u(0). Accordingly,

(P g|P h) L 2 = 1 0 (u ′ (x) -a)(P h)(x) dx = 1 0 u ′ (x)(P h)(x) dx,
since P h has mean zero by Corollary 2.2. Integration by parts yields

(P g|P h) L 2 = - 1 0 u(x)(P h) ′ (x)dx + [u(P h)] 1 0 .
By Lemma 2.1 and Corollary 2.2, we deduce that

(P g|P h) L 2 = - 1 0 u(x)h(x)dx + u(1)µ 0 (h) + (u(0) -u(1))µ 1 (h).
This shows that

-(g|h) H -1 (T ) = -(P g|P h) L 2 -µ 0 (g)µ 0 (h) = 1 0 u(x)h(x)dx -(c + u(1))µ 0 (h) + (u(1) -u(0))µ 1 (h),
as we wanted to prove.

Remark 2.12. Our Hilbert space (H, • H ) agrees with the space denoted by B 1 2 and termed the Bouziani space in [START_REF] Bouziani | On the solvability of a class of singular parabolic equations with nonlocal boundary condition in nonclassical function spaces[END_REF] and some subsequent papers by Bouziani himself and other authors, since by (2.11)

(f |g) H = 1 0 If (x)I ḡ(x)dx, ∀f, g ∈ V.

Well-posedness results

In this section we propose a general Hilbert space setting in order to study both the heat and the wave equations under (generalizations of) the integral constraints (3.1)

1 0 u(x) dx = 1 0 xu(x) dx = 0.
Namely, we take the spaces V Y equipped with the L 2 -inner product and H (resp. H -1 (T )) with the inner product in (2.13). According to Corollary 2.9, we set

H Y := H if Y = {0} 2 or Y = {0} × C, H -1 (T ) else. and therefore V Y is dense in H Y .
Our discussion is based on the weak formulation of our evolution equations, and in particular on the theory of forms. We recall that, in accordance with the terminology of [START_REF] Arendt | Semigroups and evolution equations: Functional calculus, regularity and kernel estimates[END_REF], a sesquilinear form a :

V Y × V Y → C is called H Y -elliptic (or simply elliptic) if there exist α > 0 and ω ≥ 0 such that Re a(f, f ) + ω f 2 HY ≥ α f 2 VY , ∀f ∈ V Y ; it is called coercive if it is elliptic with ω = 0; and finally it is called accretive if Re a(f, f ) ≥ 0, ∀f ∈ V Y .
We also say that it satisfies the Crouzeix estimate if for some M > 0

| Im a(f, f )| ≤ M f VY f HY , ∀f ∈ V Y .
(The name is due to the fact that forms that satisfy the Crouzeix estimate also fit the framework of [START_REF] Crouzeix | Operators with numerical range in a parabola[END_REF]).

Coming back to our setting, let Y be an arbitrary but fixed subspace of C 2 and K be a 2 × 2-matrix and consider the sesquilinear form a K defined by

(3.2) a K (f, g) := (f |g) L 2 + (KΓ 1 f |Γ 1 g) C 2 , f, g ∈ V Y , i.e., a K (f, g) = 1 0 f (x)g(x) dx + (µ 0 (f ) µ 1 (f ))K * µ 0 (g) µ 1 (g) , ∀f, g ∈ V Y ,
with form domain V Y . By Lemma 2.10 and a standard application of Young's inequality we can easily deduce H -1 (T )-ellipticity of a K . Furthermore, using Lemma 2.10 one sees that a K satisfies the Crouzeix estimate. Since V Y is dense in H Y , the Lax-Milgram Lemma yields that the form a K defined on V Y is associated with a unique linear operator (A Y,K , D(A Y,K )) defined by

D(A Y,K ) := f ∈ V Y : ∃g ∈ H Y : a K (f, h) = 1 0 (P g)(x)(P h)(x) dx + µ 0 (g)µ 0 ( h) ∀h ∈ V Y , A Y,K f := g.
Note that in the case Y = {0} 2 , the second term on the right hand side of (3.2) vanishes and hence (A Y,K , D(A Y,K )) does not really depend on K. This is why in the following we denote it simply by (A, D(A))).

D(A Y,K ) = f ∈ H 1 (0, 1) : µ 0 (f ) µ 1 (f ) ∈ Y and there exists a unique c(f ) ∈ C such that (3.3) holds , A Y,K f = -Id -1 m (f ′′ ) -c(f )δ 1 , (3.3) K µ 0 (f ) µ 1 (f ) + c(f ) + f (1) f (0) -f (1) ∈ Y ⊥ .
If Y = {0} 2 or Y = {0} × C, the same statement holds with c(f ) = 0.

Proof. We denote

K Y := f ∈ H 1 (0, 1) : µ 0 (f ) µ 1 (f )
∈ Y and there exists a unique c(f ) ∈ C such that (3.3) holds .

Let us first check the inclusion

D(A Y,K ) ⊂ K Y . Let f ∈ D(A Y,K ). Then f ∈ V Y and there exists g ∈ H -1 (T ) such that (3.4) (f |h) L 2 + (KΓ 1 f |Γ 1 h) C 2 = 1 0 (P g)(x)(P h)(x) dx + µ 0 (g)µ 0 ( h), ∀h ∈ V Y .
Because g ∈ H -1 (T ), by Corollary 2.2 P (P g) ∈ H 1 (T ) and integrating by parts we obtain

1 0 (P g)(x)(P h)(x) dx = 1 0 (P (P g)) ′ (x)(P h)(x) dx = - 1 0 (P (P g))(x)h(x) dx + [P (P g)(P h)] 1 0 , ∀h ∈ V Y .
Now, the scalar number -(KΓ 1 f |Γ 1 h) C 2 + µ 0 (g)µ 0 (h) + [P (P g)(P h)] 1 0 ∈ C is a linear combination of µ 0 ( h) and µ 1 ( h), hence it can be written in the form

c 0 µ 0 (h) + c 1 µ 1 (h) = 1 0 (c 0 + c 1 (1 -x))h(x)dx, for some c 0 , c 1 ∈ C. Letting p(x) := c 0 + c 1 (1 -x), we obtain that (f |h) L 2 = (-P (P g) + p|h) L 2 , ∀h ∈ V Y ,
where p is a polynomial of degree ≤ 1. Thus, denoting by Π the orthogonal projection of L 2 (0, 1) onto P 1 (R), we obtain (by restricting the previous identity to all h ∈ V ⊂ V Y ) (I -Π)(f + P (P g)p) = 0, or equivalently (3.5) f = (I -Π)(-P (P g) + p) + Πf = -P (P g) + Π(P (P g) + f ).

This proves that f belongs to H 1 (0, 1) and (differentiating (3.5) twice) that g = -f ′′ in the distributional sense (i.e. in D ′ (0, 1)). Hence, by (2.4), there exists c(f

) ∈ C such that A Y,K f = g = -Id -1 m (f ′′ ) -cδ 1 , and in fact c(f ) = -µ 0 (g). Note that if Y = {0} 2 or Y = {0} × C, then µ 0 (g) = 0 (as g ∈ V Y ) and hence c(f ) = 0.
It remains to check the condition (3.3). We first notice that, for all f ∈ D(A Y,K ), (3.5) leads to

f ′ = -P g + a,
for some a ∈ C. By (3.4) we obtain

1 0 f (x) h(x) dx + (KΓ 1 f |Γ 1 h) C 2 = 1 0 (-f ′ (x) + a)(P h)(x) dx -c(f )µ 0 (h), ∀h ∈ V Y .
As a 1 0 (P h)(x) dx = 0 because P h ∈ Ṽ by Corollary 2.2, we deduce that

1 0 f (x)h(x) dx + (KΓ 1 f |Γ 1 h) C 2 = - 1 0 f ′ (x)(P h)(x) dx -c(f )µ 0 (h), ∀h ∈ V Y .
By integration by parts in the first term on the right-hand side we obtain (since (P h) ′ = h by Lemma 2.1)

(KΓ 1 f |Γ 1 h) C 2 = f (0)(P h)(0) -f (1)(P h)(1) -c(f )µ 0 (h), ∀h ∈ V Y .
By Corollary 2.2 we arrive at

(KΓ 1 f |Γ 1 h) C 2 = (-c(f ) -f (1))µ 0 (h) + (f (1) -f (0))µ 1 (h), ∀h ∈ V Y .
By surjectivity of Γ 1 , cf. Remark 2.7, we have shown (3.3).

Before going on let us notice that if

Y = {0} 2 and Y = {0} × C, then (3.3) determines c(f ) uniquely. Indeed if Y = C 2 , then (3.3) is equivalent to K µ 0 (f ) µ 1 (f ) + c(f ) + f (1) f (0) -f (1) = 0, while in the case Y = Span (1, α) ⊤ with α ∈ C, then (3.3) is equivalent to K µ 0 (f ) µ 1 (f ) + c(f ) + f (1) f (0) -f (1) 1 α C 2 = 0,
which again determines c(f ) uniquely.

Let us now prove the converse inclusion. Let then f ∈ K Y . Then we can take g = -Id -1 m (f ′′ )c(f )δ 1 , with c(f ) ∈ C fixed by the condition (3.3). Moreover by Lemma 2.11, for all h ∈ L 2 (0, 1) we may write

(g|h) H -1 (T ) = - c(f ) + f (1) f (0) -f (1) µ 0 (h) µ 1 (h) C 2 + (f |h) L 2 .
But taking h ∈ V Y and using (3.3) allow us to transform the first term of this right-hand side and to obtain

(g|h) H -1 (T ) = K µ 0 (f ) µ 1 (f ) µ 0 (h) µ 1 (h) C 2 + (f |h) L 2 , ∀h ∈ V Y . This shows that a K (f, h) = (g|h) H -1 (T ) , ∀h ∈ V Y ,
and proves that f belongs to D(A Y,K ).

Remark 3.2. Let q ∈ C 1 ([0, 1]; C) with 0 < q 0 ≤ Re q(x) ≤ Q 0 for some q 0 , Q 0 ∈ R and all x ∈ [0, 1]. Then one can consider the form defined by

a K (f, g) := (qf |g) L 2 + (KΓ 1 f |Γ 1 g) C 2 , f, g ∈ V Y .
Mimicking the proof of Theorem 3.1, one can prove that the associated operator is given by

D(A Y,K ) = u ∈ H 1 (0, 1) : Γ 1 u ∈ Y , A Y,K u = -Id -1 m (qu) ′′ -c q (u)δ 1 , with c(u) = 0 if Y = {0} 2 or Y = {0} × C, otherwise c q (u) ∈ C is uniquely determined by the condition K µ 0 (u) µ 1 (u) + c q + q(1)u(1) µ 0 ((qu) ′ ) ∈ Y ⊥ .
Similar conclusions hold in the case of Theorem 5.4 below. We omit the straightforward details.

Remark 3.3. The operator (A, D(A)) associated with (a 0 , V ) is given by

D(A) = {u ∈ H 1 (0, 1) : µ 0 (u) = µ 1 (u) = 0} Au = -Id -1 m (u ′′ ).

This shows that in particular

Au = -u ′′ in D ′ (0, 1).

Remark 3.4. Contrary to the intuition, for u ∈ D(A) ⊂ H 1 (0, 1) the vector Au may not agree with -u ′′ even if u ′′ belongs to H -1 (T ). Indeed take the function u defined by

u(x) = |x - 1 2 | + αx + β,
with α, β ∈ R fixed such that µ 0 (u) = µ 1 (u) = 0. Hence we easily check that

-u ′′ = -2(δ 1 2 -δ 1 ) + αδ ′ 1 in D ′ (T ). The distribution -u ′′ cannot agree with Au since Au = -Id -1 m (u ′′ ) = -2(δ 1 2 -δ 1 ),
and elementary calculations confirm that

a(u, h) = 1 0 u(x)h(x) dx = 1 0 P (Au)(x)P h(x) dx, ∀h ∈ V.
Corollary 3.5. The operator A Y,K generates an analytic semigroup (e -tAY,K ) t≥0 of angle π 2 on H Y and also a cosine operator function with phase space V Y × H Y . It is contractive (resp., exponentially stable) if both eigenvalues of K have positive (resp., strictly positive) real part. Finally A Y,K is self-adjoint and semi-bounded if K is hermitian.

Because the embedding of H 1 (0, 1) in L 2 (0, 1) is a Hilbert-Schmidt operator (cf. [START_REF] Maurin | Abbildungen vom Hilbert-Schmidtschen Typus und ihre Anwendungen[END_REF]Satz 4]) and each e -tAY,K maps L 2 (0, 1) into H 1 (0, 1), each such operator is Hilbert-Schmidt. Recall that the composition of Hilbert-Schmidt operators is of trace class. Due to the semigroup law we hence conclude that e -2tAY,K is of trace class for all t > 0: One says that e -tAY,K t≥0 is immediately of trace class, or sometimes that it is a Gibbs semigroup.

Proof. The form a K is bounded, H Y -elliptic, and satisfies the Crouzeix estimate. It is coercive (resp., accretive) if both eigenvalues of K have strictly positive (resp., positive) real part. Hence, it follows directly from [START_REF] Crouzeix | Operators with numerical range in a parabola[END_REF]Thm. 5] that A Y,K generates a cosine operator function with associated phase space V Y × H Y and hence an analytic semigroup of angle π 2 by [3, Thm. 3.14.17]. It is known that if a bounded elliptic form is symmetric, then the associated operator is similar to its own part in the form domain, cf. [2, § 5.5.2]. Furthermore, the operator associated with a form generates three semigroups, cf. [27, Chapter 1]: one on the given Hilbert space, one on the form domain and one on the dual of the form domain (using the given Hilbert space as the pivot space). Hence we obtain the following result concerning well-posedness in a more standard L 2 -context. Theorem 3.6. Let K be hermitian and Y be a subspace of C 2 . Then the semigroup (e -tAY,K ) t≥0 on H Y leaves V Y invariant and its restriction is a semigroup on V Y that is analytic of angle π 2 and immediately of trace class.

Its generator is the part

A VY Y,K of A Y,K in V Y , which is explicitly given by D(A VY Y,K ) = u ∈ D(A Y,K ) ∩ H 2 (0, 1) : KΓ 1 (u) + Γ 2 (u) ∈ Y ⊥ , Γ 1 (u ′′ ) ∈ Y , A VY Y,K u = -u ′′ . Proof. The part of A Y,K in V Y has domain D(A VY Y,K ) := {u ∈ D(A Y,K ) : A Y,K u ∈ V Y }. But according to Theorem 3.1, for f ∈ D(A Y,K ) we have A Y,K f = -Id -1 m (f ′′ ) -cδ 1 , with c ∈ C fixed by the condition (3.3). Therefore f ′′ = -A Y,K f ∈ D ′ (0, 1), and since the condition A Y,K f ∈ V Y means in particular that A Y,K f ∈ L 2 (0, 1),
we deduce that f belongs to H 2 (0, 1).

On the other hand, using (2.12) we get

-A Y,K f = f ′′ + (f ′ (0) -f ′ (1) + c)δ 1 ∈ L 2 (0, 1). Consequently f ′ (0) -f ′ (1) + c must be zero, i.e., c = f ′ (1) -f ′ (0) = µ 0 (f ′′ ),
and A Y,K f = -f ′′ , as an equality of L 2 -functions. By (3.3) we find

KΓ 1 (u) + Γ 2 (u) ∈ Y ⊥ .
This completes the proof.

Remark 3.7. If in particular Y = {0} 2 , then each u ∈ D(A VY Y,K ) satisfies µ 0 (u ′′ ) = µ 1 (u ′′ ) = 0.
A direct computation shows that this is equivalent to the boundary conditions

u ′ (1) = u ′ (0) = u(1) -u(0).
One may extend A V {0} 2 ,K to an operator defined on the whole L 2 (0, 1) by dropping the conditions µ 0 (u) = µ 1 (u) = 0 and keeping the above boundary conditions. This new operator is perhaps more natural and has been extensively studied in [START_REF] Bobrowski | On moment preserving cosine families and semigroups in C[0, 1][END_REF].

We now obtain the following well-posedness result. It should be compared with the main result in [START_REF] Bouziani | Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions[END_REF], which our theorem below widely extends -in fact, both the allowed initial data are more general and the notion of solution is much stronger. Theorem 3.8. Let Y be a subspace of C 2 and let K be a 2 × 2-matrix.

(1) Then the heat equation

(3.6) ∂u ∂t (t, x) = ∂ 2 u ∂x 2 (t, x), t > 0, x ∈ (0, 1),
with moment conditions

(3.7) P Y ⊥ µ 0 (u(t)) µ 1 (u(t)) = 0, t > 0, and
(3.8) P Y K µ 0 (u(t)) µ 1 (u(t)) + µ 0 (u ′′ ) + u(1) u(0) -u(1) = 0, t > 0,
(here P Y and P Y ⊥ denote the orthogonal projections of C 2 onto Y and Y ⊥ , respectively) and initial condition

u(0, •) = u 0 ∈ H Y
is governed by an analytic semigroup, thus it is well-posed. If additionally K is positive definite, then

lim t→∞ u(t, •) H -1 (T ) = 0
uniformly for all initial data. (2) Similarly, the wave equation

∂ 2 u ∂t 2 (t, x) = ∂ 2 u ∂x 2 (t, x), t ≥ 0, x ∈ (0, 1),
with moment conditions (3.7)-(3.8) and initial conditions

u(0, •) = u 0 ∈ D(A VY Y,K ) along with ∂u ∂t (0, •) = u 1 ∈ D(A Y,K )
is governed by a cosine operator function, and in particular it is well-posed.

In the proof of this theorem we will need the following result.

Lemma 3.9. There holds

D(A 2 Y,K ) = {u ∈ H 3 (0, 1) : Γ 1 u, Γ 1 u ′′ ∈ Y and KΓ 1 u -Γ 2 u, KΓ 1 u ′′ -Γ 2 u ′′ ∈ Y ⊥ }.
Proof. The inclusion "⊃" is clear. In order to prove that "⊂" also holds, take

u ∈ D(A 2 Y,K ): Then u ∈ H 1 (0, 1) is such that A Y,K u = -Id -1 m u ′′ + c(u)δ 1 ∈ H 1 (0, 1). It clearly suffices to prove that u ′′ ∈ H 1 (0, 1). Now, u ′′ , v, H -1 (0,1)-H 1 0 (0,1) = Id -1 m u ′′ -c(u)δ 1 , v H-H 1 (T ) = -A Y,K u, v H-H 1 (T ) ∀v ∈ D(0, 1).
Because by assumption A Y,K u ∈ H 1 (0, 1) we deduce that in fact

u ′′ , v, H -1 (0,1)-H 1 0 (0,1) = -(A Y,K u|v) L 2 , hence u ′′ = -A Y,K u ∈ H 1 (0, 1
) and we conclude that u ∈ H 3 (0, 1), as we wanted to prove. In particular,

Id -1 m u ′′ -c(u)δ 1 = u ′′ ∈ H 1 (0, 1) by Remark 2.5, hence necessarily c(u) = 0.
Proof of Theorem 3.8. Since the operator A Y,K is a semigroup generator, well-posedness of the corresponding parabolic problem is clear. By construction this semigroup yields the solution of the evolution equation

(3.9) ∂u ∂t (t, x) = Id -1 m ∂ 2 u ∂x 2 (t, x) -c(u(t, x))δ 1 , t ≥ 0, x ∈ (0, 1),
rather than the standard heat equation. However, by standard semigroup theory we know that u(t) := e -tAY,K u 0 lies in D(A 2 Y,K ) for any t > 0, as these semigroups are analytic. By Lemma 3.9 D(A 2 Y,K ) ⊂ H 2 (0, 1) and the claim follows by Remark 2.5.

For the second assertion, since cosine functions keep the regularity of initial data but offer no additional smoothing effect, we are forced to reach H 2 (0, 1)-regularity of solutions (which by Remark 2.5 is sufficient to finally drop Id -1 m ) by actually imposing more regular data. This is obtained if u 0 lies in the domain of the generator's part in the form domain; and u 1 lies in the generator's domain, respectively.

A major feature of our semigroup approach -in particular in comparison with the Galerkin method used in some earlier articles on this subject -lies in the possibility to deduce optimal regularity results for solutions. For example it is known that due to analyticity the semigroup operators e -tAY,K map H Y into D(A Y,K ) for all Y and K for all t > 0, hence in particular they map (a closed subspace of) L 1 (0, 1) into (a closed subspace of) L ∞ (0, 1), and by the Dunford-Pettis theorem (see [2, § 7.3.1]) we deduce that each e -tAY,K is an integral operator associated with an L ∞ -kernel. Furthermore, the following holds. for all h ∈ N * . This concludes the proof, since it is well-known that an analytic semigroup maps immediately into the domain of any power of its generator.

Spectral analysis

Reminding that A Y,K has compact resolvent (due to the compact embedding of L 2 (0, 1) into H -1 (T )) and that A Y,K is self-adjoint if K is hermitian, we promptly obtain the following. Lemma 4.1. Let Y be a subspace of C 2 and let K be a 2 × 2-matrix. Then the operator A Y,K has pure point spectrum, which lies in R if K is hermitian.

In view of this Lemma, if K is hermitian we denote by λ 2 Y,K,k , k ∈ N * , the eigenvalues of A Y,K enumerated in increasing order.

In this section we will describe the spectrum of the operator A Y,K for all possible subspaces Y and when K is hermitian, obtaining in particular in all cases an asymptotic result of Weyl's type. While we do not discuss the dependence of the spectrum with respect to the variation of the subspaces Y , the spaces Y = {0} 2 and Y = C 2 represent in fact the extremal cases, in the following sense. Proposition 4.2. Let Y 1 , Y 2 be subspaces of C 2 and let K 1 , K 2 be hermitian 2 × 2-matrices. Denote by A Y1,K1 and A Y2,K2 the operators associated with the form a K1 with domain V Y1 and with the form a K2 with domain V Y2 , respectively. If Y 2 is a subspace of Y 1 and the matrix K 2 -K 1 is positive semidefinite, then

λ 2 Y1,K1,k ≤ λ 2 Y2,K2,k , ∀k ∈ N.
Proof. The assertion is a direct consequence of the Courant-Fischer minimax theorem, since the operators A Y1,K1 , A Y2,K2 are self-adjoint on Hilbert spaces (H Y1 and H Y2 respectively) that are endowed with the same norm and moreover V Y2 is a subspace of V Y1 under the above assumptions.

Combining the previous results we can slightly improve the assertion on exponential stability of (e -tAY,K ) t≥0 contained in Corollary 3.5, under the assumption that K is just positive semidefinite. Remark 4.3. Here and in the remainder of this section we repeatedly use the fact that eigenfunctions of A Y,K are in D(A 2 Y,K ) by a standard bootstrap argument, and hence in H 2 (0, 1) by Lemma 3.9. By Remark 2.5 we can hence regard them as solutions of the more usual eigenvalue problem

-u ′′ = λ 2 Y,K,k u.
Corollary 4.4. Let Y be any subspace of C 2 and K be hermitian and positive semidefinite. Then each eigenvalue of A Y,K is strictly positive, and in particular the semigroup (e -tAY,K ) t≥0 is uniformly exponentially stable.

Proof. Combining Lemma 4.1 and Proposition 4.2 we deduce that for any 1-dimensional subspace Y of C 2 the k th eigenvalue of A Y,K is always contained in the interval

[λ 2 C 2 ,K,k , λ 2 {0} 2 ,K,k ] = [λ 2 C 2 ,K,k , λ 2 {0} 2 ,0,k ] (In view of Theorem 3.1, A {0} 2 ,K = A {0} 2 ,0
). In fact, letting K be positive semidefinite we deduce -again from Proposition 4.2 -that

λ 2 C 2 ,0,k ≤ λ 2 C 2 ,K,k .
Hence, it suffices to show that all eigenvalues of A C 2 ,0 are strictly positive. First of all, A C 2 ,0 is accretive, hence all its eigenvalues are positive. To show that 0 is not an eigenvalue, and hence that λ 2 C 2 ,0,1 > 0, take u such that A C 2 ,0 u = u ′′ = 0, i. If we impose that (µ 0 (u ′′ ) + u(1), u(0)u(1)) = (0, 0), then clearly a = b = 0, i.e., u ≡ 0. This concludes the proof.

Now we look for the eigenvalues and the eigenvectors of the operator A Y,K in the hermitian case and deduce a formula of Weyl's type. Hence we can always assume that an eigenvalue λ of A Y,K is real. Let λ ∈ R and u ∈ D(A Y,K ) be such that A Y,K u = λ 2 u. Remark 4.3 yields u ′′ = -λ 2 u. By Corollary 4.4 λ = 0, and therefore u is of the form u(x) = c 1 cos(λx) + c 2 sin(λx), for some real numbers c 1 and c 2 . By direct computations we see that

µ 0 (u) = 1 0 u(x) dx = 1 λ (c 1 sin λ + (1 -cos λ)c 2 ),
and

µ 1 (u) = 1 0 (1 -x)u(x) dx = 1 λ 2 ((1 -cos λ)c 1 + (λ -sin λ)c 2 )
. For the sake of later reference we also observe that

u(1) = c 1 cos λ + c 2 sin λ, u(0) = c 1 , u ′ (1) = -λc 1 sin λ + λc 2 cos λ, u ′ (0) = λc 2 , so that (4.1) Γ 1 u = M(λ)B(λ) c 1 c 2 := 1 λ 0 0 1 λ 2 sin λ 1 -cos λ 1 -cos λ λ -sin λ c 1 c 2 while (4.2) Γ 2 u = C(λ) c 1 c 2 := λ sin λ -cos λ λ(1 -cos λ) -sin λ cos λ -1 sin λ c 1 c 2 ,
where Γ 1 , Γ 2 are the operators introduced in (2.14).

If we are imposing integral conditions associated with a general subspace Y and a general hermitian matrix K, then by Theorem 3.1 we know that the relevant conditions are

P Y ⊥ Γ 1 u = 0 and P Y (KΓ 1 u -Γ 2 u) = 0,
or rather, taking into account (4.1) and (4.2),

P Y ⊥ M(λ)B(λ) P Y (KM(λ)B(λ) -C(λ)) c 1 c 2 = 0.
Since an eigenvalue corresponds to a non-trivial solution (c 1 , c 2 ) of this linear system, we directly obtain the following result.

Theorem 4.5. Let K be a hermitian 2 × 2-matrix and Y be a subspace of C 2 . A number λ ∈ R * is such that λ 2 is an eigenvalue of A Y,K if and only if the 4 × 2-matrix

P Y ⊥ M(λ)B(λ) P Y (KM(λ)B(λ) -C(λ))
is of rank 0 or 1.

Remark 4.6. The condition in Theorem 4.5 can be specialised in an easy way.

(1) If Y = {0} 2 , then the condition in Theorem 4.5 reduces to

(4.3) D(λ) = λ sin λ + 2 cos λ -2 = 0.
In this case, each eigenvalue λ is simple and its associated eigenspace is spanned by the function

x → (sin λλ) cos(λx) + (1 -cos λ) sin(λx).

P Y u v = (x ȳ) • u v x y and P Y ⊥ u v = (-ȳ x) • u v -y x .
Consequently,

P Y u v = 0 (resp. P Y ⊥ u v = 0)
if and only if

(x ȳ) • u v = 0 (resp. (-ȳ x) • u v = 0).
This characterization and Theorem 4.5 show that λ 2 is an eigenvalue of A Y,K if and only if

(4.4) D Y,K (λ) = det (-ȳ, x)M(λ)B(λ) (x, ȳ)(KM(λ)B(λ) -C(λ)) = 0.
Corollary 4.7. Let K be hermitian and Y be a subspace of C 2 . We distinguish the following cases.

(i) If Y = {0} 2 , then (4.5) λ {0} 2 ,K,2k = 2kπ, ∀k ∈ N * , while (4.6) λ {0} 2 ,K,2k-1 = (2k -1)π + O(k -1 ), ∀k ∈ N * . (ii) If Y = C 2 , then
) λ Y,K,k = kπ + O(k -1 ), ∀k ∈ N * . (4.9
In all cases, the Weyl-type asymptotics

lim k→∞ λ 2 Y,K,k k 2 π 2 = 1 holds Proof.
We prove (i) by directly checking that 2kπ is a solution of (4.3) for all k ∈ N * . For the other roots, we notice that (4. We prove (4.6) by applying Rouché's theorem in the ball B k = B k (kπ, ǫ k ) where 0 < ǫ k ≤ 1 will be fixed later on. We first estimate |f ∞ (z)| from below on ∂B k . Suitable trigonometric formulae yield In order to prove (ii) we closely follow the arguments used to show (i). By direct calculations we see that

|f ∞ (kπ + ǫ k e it )| 2 = |f ∞ (ǫ k e it )| 2 = cosh 2 (ǫ k sin t) -cos 2 (ǫ k cos t).

Hence we have |f

∞ (kπ + ǫ k e it )| 2 ≥ cosh(ǫ k sin t) -cos(ǫ k cos t). Using the inequalities cosh(x) ≥ 1 + x 2 2 and cos(x) ≤ 1 - x 2 2 + x 4 24 , ∀x ∈ [0, 1], we obtain |f ∞ (2kπ + ǫ k e it )| 2 ≥ 1 2 ǫ 2 k - 1 24 ǫ 4 k cos 4 t ≥ 1 2 ǫ 2 k - 1
D K (λ) = 2λ(f ∞ (λ) + r(λ)),
where f ∞ (z) := 1 -cos z is analytic and the remainder r(z) := g 1 (z)z -1 + . . . + g 4 (z)z -4 (the g i being finite linear combinations of 1, cos λ, sin λ, cos 2 λ and cos λ sin λ) is also analytic (except at z = 0). The conclusion then follows by applying Rouché's theorem in the ball B k = B k (2kπ, ǫ k ) where 0 < ǫ k ≤ 1 is fixed appropriately. Finally, (iii) and (iv) follow because in view of (4.1) and (4.2), we see that

(-ȳ x)M(λ)B(λ) = 1 λ 2 (-ȳ x) λ sin λ λ(1 -cos λ) 1 -cos λ λ -sin λ = 1 λ 2 (-ȳλ sin λ + x(1 -cos λ) -ȳλ(1 -cos λ) + x(λ -sin λ)) = 1 λ 2 (-ȳλ sin λ + x(1 -cos λ) λ(x -ȳ(1 -cos λ)) -x sin λ) while (x ȳ)KM(λ)B(λ) -C(λ) = (x ȳ) k11 λ sin λ + k12 λ 2 (1 -cos λ) k11 λ (1 -cos λ) + k12 λ 2 (λ -sin λ) k21 λ sin λ + k22 λ 2 (1 -cos λ) k21 λ (1 -cos λ) + k22 λ 2 (λ -sin λ) - λ sin λ -cos λ λ(1 -cos λ) -sin λ cos λ -1 sin λ = 1 λ 2 (x ȳ) k 11 λ sin λ + k 12 (1 -cos λ) k 11 λ(1 -cos λ) + k 12 (λ -sin λ) k 21 λ sin λ + k 22 (1 -cos λ) k 21 λ(1 -cos λ) + k 22 (λ -sin λ) - λ 3 sin λ -λ 2 cos λ λ 3 (1 -cos λ) -λ 2 sin λ λ 2 (cos λ -1) λ 2 sin λ = 1 λ 2 -x(λ 3 sin λ -λ 2 cos λ) + ȳλ 2 (1 -cos λ) + q 1 (λ) -xλ 3 (1 -cos λ) + xλ 2 sin λ -ȳλ 2 sin λ + q 2 (λ) ⊤ = 1 λ 2 -xλ 3 sin λ + λ 2 (x cos λ + ȳ(1 -cos λ)) + q 1 (λ) -xλ 3 (1 -cos λ) + λ 2 sin λ(x -ȳ) + q 2 (λ) ⊤ ,
where q i (λ) is a polynomial in λ of degree ≤ 1 with coefficients which are finite linear combinations of 1, cos λ, sin λ, cos 2 λ. Therefore by (4.4) we obtain that

D Y,K (λ) = 1 λ 4 det -ȳλ sin λ + x(1 -cos λ) λ(x -ȳ(1 -cos λ)) -x sin λ -xλ 3 sin λ + λ 2 (x cos λ + ȳ(1 -cos λ)) + q 1 (λ) -xλ 3 (1 -cos λ) + λ 2 sin λ(x -ȳ) + q 2 (λ)) = x sin λ + 2λ -1 (ȳ 2 -xȳ -x2 )(1 -cos λ) + g 2 (λ)λ -2 + . . . + g 4 (λ)λ -4 ,
where the g i are finite linear combinations of 1, cos λ, sin λ, cos 2 λ, and cos λ sin λ. Hence the arguments in the proof of (ii) yield the conclusion by dividing the cases x = 0 and x = 0.

In all above cases, an eigenvalue asymptotics of Weyl's type follows directly from (4.5) and (4.6) (resp. (4.7), (4.8), (4.9)).

Remark 4.8. In the case Y = {0} 2 , using a Taylor expansion of higher order, we can even show that

λ {0} 2 ,K,2k-1 = (2k -1)π - 1 4(2k -1)π + O(k -2 ), ∀k ∈ N * .
In the same manner, in the case Y = C 2 , we see that the highest order term of the asymptotic of the eigenvalues does not depend on K, but lower order terms do. Indeed using a Taylor expansion of higher order we can even show that there exist two real numbers C 1 and C 2 (depending on K) such that

λ C 2 ,K,2k-1 = 2kπ + C 1 k + O(k -2 ), λ C 2 ,K,2k = 2kπ + C 2 k + O(k -2 ), ∀k ∈ N * .
Remark 4.9. The eigenvalue asymptotics of Weyl's type in the third and fourth case of the previous Corollary is also a consequence of the comparison principle of Proposition 4.2 and the Weyl-type formula for the first and second cases. On the other hand formulas (4.8) and (4.9) (that cannot be deduced from the comparison principle) are more precise than the formula of Weyl's type. As in Remark 4.8, the highest order term of the eigenvalue asymptotics does not depend on K, but lower order terms do.

respectively, where V Y := g y 0 ∈ V Y × C : y 0 = µ 0 (g) ≃ V Y , H 1 := g y 0 ∈ H -1 (T ) × C : y 0 = µ 0 (g) ≃ H -1 (T ).

The identifications are performed with respect to the isomorphism g µ 0 (g) → g.

In fact, V Y can be written as

V Y = x y ∈ V Y × C : Lx = y
and we also have that g (y 0 , y 1 ) ⊤ ∈ H -1 (T ) × Y : y 0 = µ 0 (g) = H 1 × {0}, or C, respectively. Also in this setting L is a bounded and surjective operator, and moreover ker L = g y 0 ∈ L 2 (0, 1) × C : y 0 = µ 0 (g), µ 1 (g) = 0 ≃ V C×{0} is dense in X 2 ≃ H -1 (T ) by Corollary 2.9.

3) Finally, let Y be the subspace spanned by a vector 1 α for some α = 0. Then g (y 0 , y 1 ) ⊤ ∈ H -1 (T ) × Y : y 0 = µ 0 (g) = g (y 0 , y 1 ) ⊤ ∈ H -1 (T ) × C 2 : y 0 = µ 0 (g), y 1 = αy 0 ≃ g y 0 ∈ H -1 (T ) × C : y 0 = µ 0 (g)

≃ H -1 (T )
whereas with the notation introduced in 2)

V Y = u φ ∈ V Y × C 2 : Γ 1 (u) = φ ∈ Y = x y ∈ V Y × C : Lx = y ≃ V Y ≃ V Y .
Since V Y is dense in H -1 (T ), the claim follows.

As in the previous sections the form a K with domain V Y is bounded and H Y -elliptic. It satisfies the Crouzeix condition. It is accretive (resp., coercive) if both eigenvalues of K have positive (resp., strictly positive) real part. It is symmetric if and only if K is hermitian. Hence we obtain the following result. Proposition 5.2. Let Y be a subspace of C 2 and let K be a 2 × 2-matrix. Then the operator -A Y,K associated with the form generates an analytic semigroup (e -tAY,K ) t≥0 of angle π 2 on H Y and also a cosine operator function with phase space V Y × H Y . This semigroup is contractive (resp., exponentially stable) if both eigenvalues of K have positive (resp., strictly positive) real part.

Again, (e -tAY,K ) t≥0 is immediately of trace class for all Y and K.

Remark 5.3. Let Y be a subspace of C 2 and let K be a hermitian 2 × 2-matrix. Then both operators A Y,K and A Y,K are self-adjoint, and it follows by a direct application of Courant's minimax formula that for all k ∈ N the k th eigenvalue of A Y,K is at least as large as the k th eigenvalue of A Y,K .

Hence, it only remains to identify the operator A Y,K . If Y = {0} 2 , then it is apparent that the above conditions reduce to those in (3.1), which have already been fully discussed in Section 3. Otherwise, the following result holds. We omit the technically involved proof, which can be performed along the lines of Theorem 3.1.

B 1 := R 0 Q 0 .
Now, Q has finite dimensional range and hence B 1 is relatively compact. Accordingly, it suffices to apply a well-known perturbation result [START_REF] Desch | Some perturbation results for analytic semigroups[END_REF] to conclude.

Remark 5.7. In the literature, inhomogeneous integral conditions of the form

1 0 u(t, x) dx = E 1 (t), 1 0
(1x)u(t, x) dx = E 2 (t), t > 0, are often considered. Reasoning as in the proof of Lemma 3.9 we see that the first coordinate of any element of D(A 2 Y ) lies in particular in H 2 (0, 1), hence the semigroup generated by A 2 Y maps immediately into functions whose first coordinate does not only satisfy the generalized heat equation (3.9), but also the classical one. In view of the known semigroup approach to evolution equations with inhomogeneous conditions (see e.

Theorem 3 . 1 .

 31 If Y = {0} 2 and Y = {0} × C, then one has

  e., u(x) := ax + b, x ∈ (0, 1), for some a, b ∈ C. Observe that µ 0 (u ′′ ) + u(1) = a + b and u(0)u(1) = -a.

( 2 )

 2 In the special case of Y = C 2 , the condition in Theorem 4.5 simplifies to requiring that λ be a root of the equation D K (λ) := det (KM(λ)B(λ) -C(λ)) = 0. (3) Similarly, if Y is spanned by (x, y) with x, y ∈ C such that |x| 2 + |y| 2 = 1, then

(4. 7 )

 7 λ C 2 ,K,2k = 2kπ + O(k -1 ) and λ C 2 ,K,2k-1 = 2kπ + O(k -1 ), ∀k ∈ N * . (iii) If Y is spanned by (0, 1), then (4.8) λ Y,K,2k = 2kπ + O(k -1 ) and λ Y,K,2k-1 = 2kπ + O(k -1 ), ∀k ∈ N * . (iv) If Y is spanned by (1, α) with α ∈ C, then

  3) is then equivalent to f ∞ (λ) + r(λ) = 0,where f ∞ (z) := sin z is analytic and the remainder r(z) := 2(cos z-1) z is also analytic (except at z = 0) and satisfies |r

where B 0 and B 1

 01 : D(A Y,K ) → H Y is the relatively compact operator defined by is a bounded linear operator on H Y defined by

  g. [3, §6.2]), Corollary 5.6 allows us to consider general inhomogeneous conditions of the formP Y ⊥ Γ 1 (u(t)) = E(t), t > 0,along with (3.7) for the inhomogeneous heat equation∂u ∂t (t, x) = ∂ 2 u ∂x 2 (t, x) + ψ(t), t > 0, x ∈ (0, 1),and to prove well-posedness of the associated evolution equation, provided some smoothness of the time dependence of ψ, E is assumed (ψ ∈ L 1 (R + , H Y ), E ∈ W 1,1 (R + , Y ⊥ ) will do for a mild solution, andψ ∈ W 1,1 (R + , H Y ), E ∈ W 2,1 (R + , Y ⊥ )will even yield a classical solution, cf. [20, §3]). In fact, in view of Corollary 4.4 the operator A Y,K is invertible as long as K is positive semidefinite and therefore we can even write down an explicit formula for the solution (in dependence of the semigroup generated by A Y,K ), cf. [20, Prop. 3.9].

  ∂B k . According to Rouché's theorem, for k large enough f admits a unique root in the ball B k , since f ∞ has this property. This proves (4.6).

							24	ǫ 4 k ≥	11 24	ǫ 2 k .
	On the other hand Hence if we take ǫ k =	C k	|r(kπ + ǫ k e it )| ≤ , with C chosen large enough that	4 kπ -1	.
	we have		11 24	ǫ k >	4 kπ -1	,	∀k ≥ 1,
			|r(z)| < |f ∞ (z)|,		∀z ∈

Acknowledgement. The first author is partially supported by the Land Baden-Württemberg in the framework of the Juniorprofessorenprogramm -research project on "Symmetry methods in quantum graphs".

This article has been partly written while the first author was visiting Valenciennes. He would like to thank the University of Valenciennes for the hospitality and for partial financial support. Both authors thank D. Mercier (Lamav, University of Valenciennes) for his help in the proof of Corollary 4.7. They are grateful to the anonymous referee for careful reading and for many useful suggestions, both on form and content.

Corollary 3.10. Let Y be a subspace of C 2 and K a 2 × 2-matrix and let u 0 ∈ H Y . Then the unique solution u to the initial value problem associated with (3.6)-(3.7)-(3.8) satisfies u(t, •) ∈ C ∞ ([0, 1]) and moreover u and its derivatives fulfill the non-local boundary conditions u (2h-1) (t, 1)u (2h-1) (t, 0) u (2h-2) (t, 1)u (2h-2) (t, 0)u (2h-1) (t, 0)

∈ Y for all h ∈ N * along with K u (2h-1) (t, 1)u (2h-1) (t, 0)

for all t > 0.

For example, for Y = {0} × C and K = 0 this amounts to saying that u(t, •) and all its derivatives fulfill periodic boundary conditions.

Proof. In a way similar to Lemma 3.9 it can be proved by induction that in fact for all h ∈ N

Observe that for u ∈ H 3 (0, 1)

Reasoning similarly we can prove by induction that in fact for u ∈ C ∞ ([0, 1])

Dynamic integral conditions

We conclude our article by discussing the heat or wave equations complemented with the condition

along with the dynamic-type one

for some subspace Y of C 2 and some 2 × 2-matrix K.

An educated guess suggests to consider the same sesquilinear form a K defined in (3.2), but now defined on

for some 2 × 2-matrix K. In fact, the following holds, where we are using the spaces H Y introduced in Section 2.

We write H := H -1 (T ) × C 2 and denote by(•|•) H the canonical inner product of the Hilbert product space H, i.e., (u|v)

Proof. The proof is performed by first considering separately the cases where Y is a Cartesian product.

1) Let us first consider the case of Y = {0} 2 or Y = {0} × C. Then the assertion can be proved letting

Observe that L : V Y → Y is a bounded and (by Remark 2.7) surjective operator. By Corollary 2.9

is dense in X 2 = H, thus the claim follows directly from [START_REF] Mugnolo | Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions[END_REF]Lemma 5.6].

2) Let now Y = C × {0} or Y = C 2 . This time, the assertion can be proved applying [START_REF] Mugnolo | Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions[END_REF]Lemma 5.6] with

and

Let Y be a subspace of C 2 , Y = {0} 2 , and let K be a 2 × 2-matrix. Then the operator matrix A Y,K associated with the form a K with domain V Y is given by

where c : H 1 (0, 1) → C is a bounded linear functional defined by (5.2)

Reasoning as in Theorem 3.6 we see that whenever

with u ∈ H 2 (0, 1), then A Y,K acts on its first component as the second derivative. Thus, the following holds.

Corollary 5.5. Let K be hermitian and Y be a subspace of C 2 . Then the semigroup (e -tAY,K ) t≥0 on H Y leaves V Y invariant and its restriction is a semigroup on V Y that is analytic of angle π 2 and immediately of trace class. Its generator is the part

Now, by standard perturbation results one may deduce that in fact our heat equation is governed by an analytic semigroup on H Y even if we replace (5.1) by the more general condition

for any bounded linear operator Q : H 1 (0, 1) → Y .


Bart Dioos
email: bart.dioos@wis.kuleuven.

AND Joeri Van Der Veken
email: joeri.vanderveken@wis.kuleuven.be

LAMAV Luc Vrancken
email: luc.vrancken@univ-valenciennes.fr

SEQUENCES OF HARMONIC MAPS IN THE 3-SPHERE

Keywords: 2010 Mathematics Subject Classification. Primary 58E20; Secondary 53C42 3-sphere, almost complex surface, harmonic map, pseudo-holomorphic curve, quaternions

We define two transforms between non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between non-conformal harmonic maps into the 3sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S 3 × S 3 . As a consequence we can construct sequences of H-surfaces and almost complex surfaces.

Introduction

Consider a map f : S → S 3 from a Riemann surface S into the unit 3-sphere. The map f is harmonic if it satisfies the equation ∆f + |df | 2 f = 0 where ∆ is the Laplacian on the surface S ( [START_REF] Hélein | Harmonic maps, conservation laws and moving frames[END_REF]). The map f is conformal if it preserves the conformal structure on S, that is, ∂f, ∂f = 0 where ∂ stands for ∂ ∂z . In this case, f is a minimal immersion of the surface in the 3-sphere. For a recent and broad survey on harmonic maps, the reader is referred to [START_REF] Hélein | Handbook of global analysis[END_REF]. In this article we will almost always assume that f is not conformal: ∂f, ∂f = 0. We will show that to a non-conformal harmonic map from S to S 3 one can associate two new maps from S to S 3 which are also non-conformal and harmonic. In fact one can define a sequence {f p | p ∈ Z} of non-conformal harmonic maps from S into S 3 where f 0 = f . These will be the main results of Section 3.

The transforms of harmonic maps were inspired by the work [START_REF] Bolton | Transforms for minimal surfaces in the 5-sphere[END_REF]. In that article Bolton and the last author described transforms to minimal surfaces in S 5 with non-circular ellipse of curvature. Antic and the last author [START_REF] Antic | Sequences of minimal surfaces in S 2n+1[END_REF] generalized these transforms for superconformal minimal surfaces in odd-dimensional spheres S 2n+1 whose (n -2) higher-order ellipses of curvature are circles. These transforms are natural generalisations of the polar construction for superconformal minimal surfaces in odd-dimensional spheres (see [START_REF] Bolton | Minimal surfaces and the affine Toda field model[END_REF] and for surfaces in S 3 see [START_REF] Lawson | Complete minimal surfaces in S 3[END_REF]).

In Section 2 we first give some preliminaries on quaternions. Quaternions turn out to be very convenient for describing surfaces in the 3-sphere. A study of surfaces in spheres using a quaternionic language can be found in [START_REF] Burstall | Conformal geometry of surfaces in S 4 and quaternions[END_REF]. In this reference however conformal maps are studied whereas in this article we will mostly consider non-conformal maps. Next we will define an adapted frame for non-conformal harmonic maps and derive the moving frame equations and the compatibilty conditions for this frame.

In Section 4 we will address the following problem. Given a non-conformal harmonic map f from a surface into the 3-sphere we can transform it into a new non-conformal harmonic map. When are a non-conformal map f and its transformed map equal up to an isometry of S 3 ? The maps satisfying this property are certain parametrizations of Clifford tori. Since the transforms are generalizations of Lawson's polar construction for minimal surfaces, this question is analogue to the question when a minimal surface is congruent to its polar surface. The only minimal surface in S 3 congruent to its polar is the Clifford torus S 1 ( 1

√ 2 ) × S 1 ( 1 √ 2 ).
In order to answer our question we will prove an existence and uniqueness theorem (Proposition 4.1) that very much resembles the classical existence and uniqueness theorem of Bonnet. Using the Bonnet-type theorem and a technical lemma, we can answer our question in Theorem 4.3.

The original motivation of the authors to investigate these transforms of harmonic maps is the study of almost complex surfaces in the nearly Kähler manifold S 3 × S 3 . In the paper [START_REF] Bolton | Almost complex surfaces in the nearly Kähler S 3 × S 3[END_REF], Bolton, Dillen, Dioos and Vrancken found a correspondence between almost complex surfaces in S 3 × S 3 and H-surfaces X in R 3 , that is, surfaces that satisfy the Wente H-equation

X xx + X yy = -4 √ 3 X x × X y .
Moreover, on such an almost complex surface there exists a quadratic holomorphic differential. In Section 5 of the present paper we will discuss the relation between harmonic maps to S 3 , H-surfaces in R 3 and almost complex surfaces in S 3 × S 3 . In Theorem 5.3 we prove that almost complex surfaces in S 3 × S 3 correspond to harmonic maps in the 3-sphere, and vice versa. This correspondence follows quickly from the results in [START_REF] Bolton | Almost complex surfaces in the nearly Kähler S 3 × S 3[END_REF] we just mentioned and a non-conformal analogue of Lawson's correspondence Theorem (Proposition 5.2). As a corollary one can associate a whole sequence of such surfaces to one given almost complex surface in S 3 × S 3 with non-vanishing differential.

Harmonic maps to S 3

The ring of quaternions H can be identified with the vector space R 4 . If quaternions are written as real linear combinations of the basis elements 1, e 1 , e 2 and e 3 , then the quaternion multiplication is determined completely by the identities e 2 1 = e 2 2 = e 2 3 = e 1 e 2 e 3 = -1. A quaternion that is a linear combination of e 1 , e 2 and e 3 is called an imaginary quaternion. The set of imaginary quaternions Im H can be identified with the Euclidean space R 3 . The product of two imaginary quaternions α and β is given by

(1) αβ = -α, β + α × β
where , is the Euclidean inner product and × is the usual vector product on R 3 . The quaternions are very useful to describe the 3-sphere and its tangent spaces. The 3-sphere S 3 is the set of unit quaternions {p ∈ H | p = 1}. One can prove that uv, uw = u, u v, w for all quaternions u, v and w. Therefore pα is orthogonal to p for every imaginary quaternion α and the tangent space at p is (2)

T p S 3 = {pα | α ∈ Im H}.
On a surface we will use complex coordinates, so in order to describe the complexified tangent vectors we will need the complexified quaternions

H ⊗ C = H ⊕ iH.
The element i must be distinguished from e 1 ∈ H. The complex bilinear extension of the Euclidean metric and vector product will also be denoted by , and ×. The product of two complexified quaternions p 1 + ip 2 and q 1 + iq 2 is

(p 1 + ip 2 )(q 1 + iq 2 ) = (p 1 q 1 -p 2 q 2 ) + i(p 1 q 2 + p 2 q 1 ).
The conjugate of a complexified quaternion p 1 + ip 2 is equal to

p 1 + ip 2 = p 1 -ip 2
and is denoted with a bar. In contrast, the conjugate of a quaternion p is written as

p * = (a + be 1 + ce 2 + de 3 ) * = a -be 1 -ce 2 -de 3 .
So ¯means conjugation with respect to the imaginary i and * is conjugation with respect to the three imaginaries e 1 , e 2 , e 3 . Now consider a harmonic map f : S → S 3 ⊂ H from a Riemann surface S into the 3-sphere S 3 . Choose a local complex coordinate z = x + iy on S. Denote ∂ ∂z and ∂ ∂ z by ∂ and ∂ respectively. Similarly the derivatives of f with respect to the real coordinates x and y will be written as f x and f y respectively. We introduce the H ⊗ C-valued function

f 1 = ∂f. Since f, f = 1, it follows that f, f 1 = 0.
Harmonicity means that ∂ ∂f = -|∂f | 2 f . If f is in addition a conformal map, f is a minimal isometric immersion of S in S 3 . We assume that f is not conformal. By the harmonicity of f , the function f 1 , f 1 is holomorphic. Since f is assumed to be non-conformal, f 1 , f 1 is non-zero. Therefore there exists a complex coordinate z such that f 1 , f 1 = -1. We will call such a coordinate an adapted complex coordinate for f . By (2) there exist functions α and β with values in Im H such that f x = f α and f y = f β. This means that

f 1 = 1 2 f (α -iβ). It follows from f 1 , f 1 = -1 that α, α -β, β = -4, α, β = 0.
Hence there is a non-negative smooth function φ such that

|α| = 2 sinh φ, |β| = 2 cosh φ. (3)
Note that β is nowhere vanishing by equation ( 3), but α can be zero. At points were α is not zero the vectors f 1 and f1 are linearly independent and φ is positive. In the following we will tacitly assume that we are working on the open subset U of S where α = 0. On this set φ is positive and the image f (U ) is a surface in the 3-sphere. At a point of U define N to be the real unit vector in the direction of f (α × β). Then N is orthogonal to {f, f 1 , f1 } and

|f (α × β)| 2 = α, α β, β -α, β 2 = 4 sinh 2 2φ, hence f (α×β) = ±2 sinh 2φ N . For definiteness we take N such that the orthogonal frame {f, f x , f y , N } in R 4 is positively oriented, so N = 1 2 csch 2φ f (α × β).
We have now defined a complex moving frame F = {f, f 1 , f1 , N } for f on the set U . The matrix A of complex inner products of the vectors of F is

(4) A =     1 0 0 0 0 -1 cosh 2φ 0 0 cosh 2φ -1 0 0 0 0 1     .
Now we give the moving frame equations for F . Write µ = ∂f 1 , N . An easy calculation using (4) gives the equations in terms of φ and µ.

∂f = f 1 , ∂f 1 = f + 2∂φ(coth 2φ f 1 + csch 2φ f1 ) + µ N, ∂ f1 = -cosh 2φ f, ∂N = -µ csch 2φ(csch 2φ f 1 + coth 2φ f1 ). ( 5
)
The corresponding ∂-equations can be found by simply taking the conjugates of the above ones. The compatibility conditions ∂ ∂F = ∂∂F for the frame F give 2∂ ∂φ = -sinh 2φ + |µ| 2 csch 2φ, ∂µ = -2μ∂φ csch 2φ. [START_REF] Burstall | Conformal geometry of surfaces in S 4 and quaternions[END_REF] The complex function µ measures the rate at which the image of f is pulling away from the great 2-sphere tangent to the image of f . If f is a map into a great 2sphere, then µ vanishes and the above compatibility condition for φ becomes the sinh-Gordon equation.

Remarks 2.1.

(1) A non-conformal harmonic map f from a compact surface S to the 2-sphere always must have a singular point. Indeed, suppose such a map does not have singular points, then φ is a positive function on a compact surface satisfying the sinh-Gordon equation. But then the maximum principle gives a contradiction.

(2) Recall that by Hopf's lemma there are no non-conformal harmonic maps from the 2-sphere to the 3-sphere.

The transforms

Let f : S → S 3 be a non-conformal harmonic map. In this section we will first show how to associate to f two new non-conformal harmonic maps f + and f - from S into S 3 . Moreover, if z is an adapted complex coordinate for the map f , then it also is an adapted coordinate for f + and f -. Then we will show that the two transformations are each others inverse in the sense that (f + ) -= (f -) + = f . Consequently we can associate a sequence {f p | p ∈ Z} of non-conformal harmonic maps from S into S 3 to the map f 0 = f . Fix a point p ∈ S 3 and consider the vectors

± sin θ f β |f β| + cos θN (7) in T f (p) S 3
, where θ is chosen such that cos θ = |α|/|β| = tanh φ and sin θ = sech φ.

The ellipse E with f α and f β as minor and major semi-axes is the image of a circle in the tangent plane to S at p under df . The cosine cos θ is the ratio between the lengths of the minor and major axes of this ellipse and is a measure for its eccentricity as well as for the non-conformality of f . The vectors above now have the following geometrical meaning. Let R θ be the rotation of T f (p) S 3 about the minor axis of E through the angle θ. Then the orthogonal projection of the rotated ellipse R θ (E) onto the plane containing E is a circle. Of course, the same holds for the rotation R -θ = R -1 θ . The vectors above are the images of the unit normal N under the rotations R θ and R -θ .

We can rewrite the vectors in [START_REF] Hélein | Harmonic maps, conservation laws and moving frames[END_REF] as

f + = i 2 sech 2 φ(f 1 -f1 ) + tanh φ N, f -= - i 2 sech 2 φ(f 1 -f1 ) + tanh φ N, (8)
where we have used f β = i(f 1 -f1 ) and |f β| = 2 cosh φ. By varying the point p, we can look at f + and f -as maps from S to S 3 again and we call them the (+)transform and (-)transform of f respectively. If f were conformal, that is if |α| = |β| everywhere, the expressions (7) would still make sense. In fact, we get θ = 0 and recover the polar surface of the minimal surface f , see [START_REF] Lawson | Complete minimal surfaces in S 3[END_REF]. From now on we restrict to the non-conformal case and we denote the transforms ( 8) by f ε where ε is 1 or -1.

All objects and functions related to f ε will be denoted with a superscript ε. For instance, α ε and β ε are the functions such that ∂f ε = 1 2 f ε (α ε -iβ ε ) and φ ε is the non-negative smooth function that satisfies the ε-analogue of (3). Remark 3.1. In [7, p. 64-65] Hélein describes how to associate to a harmonic map from a surface to S 2 two new harmonic maps from the surface into the 3-sphere. These harmonic maps are conformal, so they are different from the transforms we describe here. Theorem 3.2. Let f : S → S 3 be a non-conformal harmonic map from a Riemann surface S into the 3-sphere. Then the tranforms f + and f -are also non-conformal harmonic maps from S to S 3 . Furthermore, an adapted complex coordinate for f is also an adapted complex coordinate for f + and f -.

Proof. First we define f ε 1 = ∂f ε . Using the definition (8) of f ε and the moving frame equations ( 5) for F one gets ( 9)

f ε 1 = εif - 1 2 (µ -2εi∂φ) sech 2 φ(csch 2φ f 1 + coth 2φ f1 -εi N ).
A computation using the inner products (4) then gives

f ε 1 , f ε 1 = -1, |f ε 1 | 2 = 1 + 1 2 |µ -2εi∂φ| 2 sech 2 φ. (10)
So the coordinate z is an adapted complex coordinate for f ε as well.

Next we will show that f ε is a non-conformal harmonic map. We have just showed that f ε 1 , f ε 1 is non-zero, so f ε 1 clearly is non-conformal. We still have to check that ∂ ∂f ε is a multiple of f ε . An easy calculation using (4) gives

f ε , f = 0, f ε , f 1 = -εi, f ε , f1 = εi, f ε , N = tanh φ. (11) Equation (9) gives f ε 1 , f = εi, f ε 1 , f 1 = -tanh φ(µ -2εi∂φ), f ε 1 , f1 = 0, f ε 1 , N = εi 2 sech 2 φ(µ -2εi∂φ). ( 12
)
Now we can calculate the inner products of ∂ ∂f ε with the frame vectors of F . The first inner product is

∂ ∂f ε , f = ∂ f ε 1 , f -f ε 1 , f1 = 0.
With a similar calculation using the compatibility conditions (6) and equation [START_REF] Wente | Explicit solutions to the H-surface equation[END_REF] the other inner products become

∂ ∂f ε , f 1 = εi|f ε 1 | 2 , ∂ ∂f ε , f1 = -εi|f ε 1 | 2 , ∂ ∂f ε , N = -tanh φ|f ε 1 | 2 .
Comparing with (11) we have now shown that ∂ ∂f ε = -|f ε 1 | 2 f ε and thus f ε is an harmonic map into the 3-sphere. Proof. From the expressions for f ε 1 , f ε 1 and |f ε 1 | 2 it follows that α ε , α ε = |µ -2εi∂φ| 2 sech 2 φ. So φ ε is non-negative and is zero if and only if µ = 2εi∂φ. The compatibility conditions [START_REF] Burstall | Conformal geometry of surfaces in S 4 and quaternions[END_REF] for the moving frame F give ∂(µ -2εi∂φ) = εi sinh 2φ -μ csch 2φ(µ -2εi∂φ) .

If µ = 2εi∂φ on an open set, the equation gives sinh 2φ = 0, which means φ = 0. This gives a contradiction.

By Lemma 3.3 the normal N ε and the moving frame

F ε = {f ε , f ε 1 , f ε 1 , N ε }
of the new harmonic map f ε are well-defined on an open dense set U ε . The ε-analogues of the moving frame equations and compatibility conditions for F ε hold on U ε .

Next we prove that the (+)transform and (-)transform are mutual inverses. Before doing so, we calculate the functions α ε , β ε and α ε × β ε in terms of α, β, φ and µ. Lemma 3.4. If µ = µ 1 + iµ 2 and φ x = ∂φ ∂x and φ x = ∂φ ∂y , then

α ε = (µ 2 -εφ x ) csch 2φ α + 1 2 (µ 1 -εφ y ) tanh φ sech 2 φ β - ε 2 csch 2 φ α × β , β ε = (µ 1 -εφ y ) csch 2φ α + sech 2 φ 1 -1 2 (µ 2 -εφ x ) tanh φ β + ε 2 sech 2 φ 1 + (µ 2 -εφ x ) csch 2φ α × β, α ε × β ε = 2ε csch 2φ(µ 1 -εφ y ) α - ε 2 sech 4 φ |µ| 2 + (µ 2 -εφ x ) sinh 2φ β - 1 4 sech 4 φ |µ| 2 + 2(µ 2 -εφ x ) coth φ α × β.
Proof. The ε-transform f ε is a map to S 3 , so

f ε (f ε ) * = 1. Hence α ε -iβ ε = 2(f ε ) * f ε 1 .
The definition of the transform f ε and expression [START_REF] Lawson | Complete minimal surfaces in S 3[END_REF] for

f ε 1 give α ε -iβ ε = εi sech 2 φ(f 1 -f1 ) * + 2 tanh φ N * • εif - 1 2 (µ -2εi∂φ) sech 2 φ(csch 2φ f 1 + coth 2φ f1 -εi N
The equality (pα) * = -αp * holds for every quaternion p and every imaginary quaternion α. This equality and (3) give

N * f = - 1 2 csch 2φ α × β, (f 1 -f1 ) * f = iβ, N * f 1 = - 1 2 tanh φ β + i coth φ α , (f 1 -f1 ) * f 1 = -2 cosh 2 φ - i 2 α × β, N * f1 = - 1 2 tanh φ β -i coth φ α , (f 1 -f1 ) * N = i coth φ α.
Substitution of these equations give

α ε -iβ ε = -i sech 2 φ β + ε 2 α × β -(µ -2εi∂φ) i csch 2φ α -1 2 tanh φ sech 2 φ β + ε 4 coth φ sech 4 φ α × β .
After expanding the product in the last term and simplifying some hyperbolic trigonometric identities, one obtains the expressions for α ε and β ε . The expression for α ε × β ε can be calculated by using the formula u × (v × w) = u, w v -u, v w and the inproducts (3). Theorem 3.5. Let f : S → S 3 be a non-conformal harmonic map. Then the (+)transform and (-)transform of f are mutual inverses in the sense that

(f + ) -= (f -) + = f.
Proof. Consider a non-conformal harmonic map f : S → S 3 . Let ε and ε be such that εε = -1. Note that

f ε = 1 2 sech 2 φf εβ + 1 2 α × β . Therefore (f ε ) ε = 1 2 sech 2 φ ε f ε εβ ε + 1 2 α ε × β ε = 1 4 sech 2 φ sech 2 φ ε f εβ + 1 2 α × β εβ ε + 1 2 α ε × β ε .
Lemma 3.4 and a good amount of arithmetic then yield

(13) (f ε ) ε = 1 4 sech 2 φ sech 2 φ ε f |µ -2εi∂φ| 2 + 4 cosh 2 φ = f.
So the (+)transform and (-)transform are each others inverses.

Theorem 3.5 allows us to associate to a non-conformal harmonic map f : S → S 3 a sequence {f p | p ∈ Z} of such harmonic maps by defining f 0 = f and, for every integer p, f p+1 = (f p ) + and f p-1 = (f p ) -. Moreover, if z is an adapted complex coordinate for one of the maps f p in the sequence, then it is an adapted complex coordinate for every map in the sequence. In the rest of the article we will refer to this sequence as the sequence associated to the map f .

Sequences of harmonic maps

In this section we answer the following question: when is a non-conformal harmonic map f from a surface to the 3-sphere the same as its ε-transform f ε ? Two maps are considered to be the same if they are equal up to an isometry of the 3-sphere. Note that in this case all maps in the sequence associated to f are congruent by Theorem 3.5. Proposition 4.1 gives a criterium for this. It tells us that two non-conformal harmonic maps are equal, up to an isometry of S 3 , if their respective functions φ and µ as defined in Section 2 agree. Proposition 4.1 is an analogue of the classical existence and uniqueness theorem of Bonnet. Next we find two relations between the functions φ ε and µ ε of the ε-transform f ε and φ and µ of the original map f . With the use of the Bonnet-type theorem and the relations between φ, µ and φ ε , µ ε , we determine all the non-conformal maps that are congruent to their ε-transforms. Proposition 4.1. Let φ be a real positive smooth function and µ a complex function on a simply connected surface S satisfying the differential equations [START_REF] Burstall | Conformal geometry of surfaces in S 4 and quaternions[END_REF]. Then there exists a non-conformal harmonic map f : S → S 3 such that all the inner products of the frame F = {f, f 1 , f1 , N } are given by (4) and ∂f 1 , N = µ. Furthermore such a map f is unique up to isometries of S 3 : if g is another map satisfying the above conditions, then g = R • f where R ∈ O(4).

Proof. (Existence.) Fix a point p in S. We consider the moving frame equations (5) as a system of first order differential equations with the R 4 -valued functions f , f 1 , f1 and N as variables. Since the integrability conditions are assumed to hold, there exists a unique solution f , f 1 , f1 and N on S such that the ten inner products f, f , f, f 1 , . . . , N, N are given by the matrix (4) at the point p.

The inner products satisfy the system of twenty linear differential equations

∂ f, f = F 1 ( f, f , . . . , N, N , φ, µ), ∂ f, f = F ′ 1 ( f, f , . . . , N, N , φ, µ), . . . . . . ∂ N, N = F 10 ( f, f , . . . , N, N , φ, µ), ∂ N, N = F ′ 10 ( f, f , . . . , N, N , φ, µ).
The functions F 1 , . . . , F ′ 10 are linear with respect to the inner products and depend on φ and µ. It is a straightforward task to check that the functions in the entries of the matrix A also satisfy this system of equations. Since both solutions agree at the point p, they must be the same at all points of S.

Finally note that the two differential equations ∂f = f 1 , ∂f = f1 are integrable, so there exists a smooth map f from S to R 4 . The inner products f, f 1 and f, f1 are zero, so f is a map into a 3-sphere. The first and third moving frame equations imply that f is harmonic. The non-conformality is clear. The existence part is now proven.

(Uniqueness.) Assume that there are two non-conformal harmonic maps f and g from S into S 3 such that φ f = φ g and µ f = µ g . After applying an isometry of the 3-sphere we may assume that the tangent vectors and normals of f and g agree p:

f (p) = g(p), f 1 (p) = g 1 (p), N f (p) = N g (p).
Moreover by the assumption the maps f and g satisfy the same moving frame equations [START_REF] Dioos | Flat almost complex surfaces in S 3 × S 3[END_REF]. By the uniqueness for solutions of ordinary differential equations f and g are the same.

This proposition will allow us to give an alternative proof of two uniqueness and existence theorems in [START_REF] Dioos | Flat almost complex surfaces in S 3 × S 3[END_REF]. We come back to this remark in Remark 5.4 (2). Lemma 4.2. Let f : S → S 3 be a non-conformal harmonic map and f ε its εtransform. Then the functions φ, µ and φ ε , µ ε satisfy the following relations.

4 sinh 2 φ ε = |µ -2εi∂φ| 2 sech 2 φ (14) 4 sinh 2 φ = |µ ε + 2εi∂φ ε | 2 sech 2 φ ε (15) tanh φ ε (µ ε + 2εi∂φ ε ) = tanh φ(µ -2εi∂φ) (16)
Proof. Equation ( 14) was already found in the proof of Lemma 3.3. By Theorem 3.5 we know that (φ ε ) ε = φ for ε = -ε, so equation (15) follows from (14):

4 sinh 2 φ = 4 sinh 2 (φ ε ) ε = |µ ε + 2εi∂φ ε | 2 sech φ ε .
For equation ( 16) we consider the matrices M and M ε such that

  α ε β ε α ε × β ε   = M ε 0   α β α × β   ,   (α ε ) ε (β ε ) ε (α ε ) ε × (β ε ) ε  = M ε ε   α ε β ε α ε × β ε   .
The first equation is just the matrix notation of the three equations in Lemma 3.4, so the elements of M ε 0 can readily be read from those equations. The expression for the matrix M ε ε is the same as the one for M ε 0 , but one has to perform the changes ε → -ε, φ → φ ε and µ → µ ε . By Theorem 3.5 we know that M ε ε M ε 0 = I. The equations in the (1, 1)-and the (2, 1)-entry of

M ε ε M ε 0 = I give us the equations (µ 1 -εφ y )(µ ε 1 + εφ ε y ) + (µ 2 -εφ x )(µ ε 2 + εφ ε x ) = sinh 2φ sinh 2φ ε , (µ 2 -εφ x )(µ ε 1 + εφ ε y ) -(µ 1 -εφ y )(µ ε 2 + εφ ε x ) = 0. ( 17
)
The diligent reader can check that the other seven equations in M ε ε M ε 0 = I become trivial after substitution of ( 14), (15) and the equations (17). The system of equations ( 17) is linear in µ ε 1 + εφ ε y and µ ε 2 + εφ ε x and its determinant is -|µ -2εi∂φ| 2 = -4 sinh 2 φ ε cosh 2 φ, which by Lemma 3.3 is non-zero on the open dense subset U ε . Solving the system of equations gives

µ ε 1 + εφ ε y = tanh φ coth φ ε (µ 1 -εφ y ), µ ε 2 + εφ ε x = tanh φ coth φ ε (µ 2 -εφ x )
, which are the real and the imaginary part of equation ( 16). Now we prove the equivalence of (b) and (c). If φ = φ ε and µ = µ ε , then it follows from equation ( 16) that ∂φ is zero, so φ is constant. This proves one implication. For the converse implication the equations ( 14) and ( 6) give 4 sinh 2 φ ε = |µ| 2 sech 2 φ = 4 sinh 2 φ, so φ = φ ε and therefore µ = µ ε by ( 16).

Next we prove the equivalence of (c) and (d). If φ is constant, the two compatibility conditions (6) give |µ| 2 = sinh 2 2φ and ∂µ = 0. So µ is a holomorphic function with constant modulus. By the maximum modulus principle µ is a constant function. Since φ is positive, µ is non-zero. Conversely if µ is a non-zero constant then φ also is constant by the second condition in ( 6) and the positiveness of φ.

The last assertion follows directly from the fact that the (+) and (-)transforms are each others inverse.

In order to find explicit expressions for the maps that satisfy the conditions in the previous theorem one can in principle integrate the moving frame equations [START_REF] Dioos | Flat almost complex surfaces in S 3 × S 3[END_REF]. This approach however is not very practical. Therefore we will follow another approach. Note that the images of the harmonic maps with constant φ and µ are surfaces with constant principal curvatures. Furthermore these surfaces are flat, because the frame vectors f , f x , f y and N are orthogonal and have constant length. Therefore the harmonic maps with constant φ and µ must be certain reparametrizations of Clifford tori, because these tori are the only flat surfaces with constant principal curvatures in S 3 . This observation leads to the next classification.

Theorem 4.4. Consider the maps f : R 2 → S 3 defined by (18) f (x, y) = r cos(ax + by), r sin(ax + by), s cos(cx + dy), s sin(cx + dy) , where r and s are non-zero real constants satisfying r 2 + s 2 = 1 and

a = 2 r sinh φ cos θ, b = - 2 r cosh φ sin θ, c = 2 s sinh φ sin θ, d = 2 s cosh φ cos θ (19)
for some real constants φ > 0 and θ related by

θ = 1 2 arccos (s 2 -r 2 ) cosh 2φ . ( 20
)
These maps are non-conformal harmonic maps with adapted coordinate z = x + iy and the constant φ plays the role of the function φ in the moving frame equations [START_REF] Dioos | Flat almost complex surfaces in S 3 × S 3[END_REF], in particular f x , f x = 4 sinh 2 φ and f y , f y = 4 cosh 2 φ are constant. The function µ appearing in (5) is also constant, more precisely,

(21) µ = sinh 2φ 2rs (r 2 -s 2 ) sinh 2φ -i 1 -(r 2 -s 2 ) 2 cosh 2 2φ .
Moreover, these maps are congruent to their ε-transforms and, conversely, all nonconformal harmonic maps congruent to their ε-transform are of the form (18).

This gives (b). The compatibility conditions (6) for µ and µ ε give ∂(µ -µ ε ) = -2(μ -με )∂φ csch 2φ.

After substitution of equation (a) this becomes

(22) ∂ ∂φ = 2|∂φ| 2 csch 2φ.
Substituting the compatibility conditions (6) for φ then gives equation (c). Deriving the function coth φ ∂φ and using equation ( 22) directly shows that the function is holomorphic.

Lemma 4.6. Assume that µ = µ ε . (a) If µ is zero, then cosh φ cosh φ ε = c for some constant c. (b) If µ is non-zero, then φ = φ ε and f is congruent to f ε .
Proof. If we assume that µ is zero, then (16) gives ∂φ tanh φ = -∂φ ε tanh φ ε , which after integration becomes ln(cosh φ cosh φ ε ) = ln c where c is a positive constant. We note that taking the ∂-derivative of the first mentioned equation does not yield any new information. Now assume that µ does not vanish. The second compatibility condition of ( 6)

gives 0 = ∂(µ -µ ε ) = -2μ(∂φ csch 2φ -∂φ ε csch 2φ ε ), hence (23) ∂φ csch 2φ = ∂φ ε csch 2φ ε .
Deriving this equation with respect to ∂ and using the first compatibility condition of (6) yields

|µ| 2 (csch 2 2φ -csch 2 2φ ε ) = 4(|∂φ| 2 coth 2φ csch 2φ -|∂φ ε | 2 coth 2φ ε csch 2φ ε ).
After substitution of (23) this equation becomes

|µ| 2 (csch 2 2φ -csch 2 2φ ε ) = 4|∂φ| 2 csch 2 2φ(cosh 2φ -cosh 2φ ε ).
Now note that the left and right hand side have opposite signs, so both sides have to vanish. Therefore, since µ is non-zero, it follows that φ = φ ε . By Theorem 4.3 (b) the map f and its ε-transform are congruent.

5.

Almost complex surfaces in S 3 × S 3

In this section we discuss the relation between harmonic maps f : S → S 3 and almost complex surfaces in the nearly Kähler manifold S 3 × S 3 . Before showing this relation, we first briefly recall some definitions and the necessary background on almost complex surfaces in S 3 ×S 3 . For more details the reader is referred to [START_REF] Bolton | Almost complex surfaces in the nearly Kähler S 3 × S 3[END_REF].

An almost Hermitian manifold is a manifold endowed with an almost complex structure J and a Riemannian metric that is compatible with J. If in addition the tensor field ∇J is skew-symmetric, where ∇ is the Levi-Civita connection of the metric, then the manifold is called nearly Kähler. The product manifold S 3 × S 3 of two 3-spheres admits such a nearly Kähler structure. The almost complex structure J on S 3 × S 3 is defined by

J(X, Y ) (p,q) = 1 √ 3 2pq -1 Y -X, -2qp -1 X + Y
for (X, Y ) ∈ T (p,q) S 3 × S 3 . It is easy to check that J is anti-involutive. If we denote the usual product metric on S 3 × S 3 by , , then the nearly Kähler metric g is given by

g(Z, W ) = 1 2 Z, W + JZ, JW
and it is easily seen that g is compatible with J. Almost complex surfaces in S 3 ×S 3 , also known as pseudo-holomorphic curves, are surfaces for which the almost complex structure J maps tangent vectors onto tangent vectors. On S 3 × S 3 there is also an almost product structure P (i.e. an involutive endomorphism) defined by

P (X, Y ) (p,q) = (pq -1 Y, qp -1 X).
If ψ : S → S 3 × S 3 is an almost complex surface and z a complex coordinate, then g(P ψ z , ψ z ) dz 2 defines a holomorphic quadratic differential on the surface. A H-surface in the Euclidean 3-space is a surface X satisfying the equation

(24) X xx + X yy = 2HX x × X y
where z = x + iy is a complex coordinate and H is a real function. In this article H always is constant. It is important to note that the definition of a H-surface does not depend on the choice of complex coordinates. To show this we follow [START_REF] Wente | Explicit solutions to the H-surface equation[END_REF]. The solutions of (24) are the critical maps of the functional E H (X) = D(X)+ 4HV (X).

Here D(X) is the Dirichlet energy functional

D(X) = S |dX| 2 dxdy and V (X) is the volume integral V (X) = 1 3 S X, X x × X y dxdy.
The functional E H is invariant under changes of complex coordinates, so we may change the complex coordinate on the H-surface. Furthermore note that the quadratic differential X z , X z dz 2 is holomorphic. If X z , X z dz 2 vanishes, X is a surface of constant mean curvature H. We want to warn the reader that by a H-surface many authors mean a constant mean curvature surface. We will not follow this terminology; instead we follow Wente [START_REF] Wente | Explicit solutions to the H-surface equation[END_REF]. So in this article a H-surface is not necessarily a constant mean curvature surface and the differential X z , X z dz 2 must not be zero.

The results of [START_REF] Bolton | Almost complex surfaces in the nearly Kähler S 3 × S 3[END_REF] we need in this section will be summarised in the following theorem.

Theorem 5.1 ([2]

). Let ψ : S → S 3 × S 3 be a simply connected almost complex surface and z = x + iy a complex coordinate. To such an almost complex surface one can associate a surface X in Euclidean 3-space satisfying the H-equation

(25) X xx + X yy = - 4 √ 3 X x × X y ,
and vice versa. Two almost complex surfaces are congruent in S 3 × S 3 if and only if their corresponding surfaces in R 3 are congruent. Moreover, the holomorphic differential on the almost complex surface satisfies g(P ψ z , ψ z ) = e i π 3 X z , X z . Thus an almost complex surface with vanishing holomorphic differential corresponds to a surface in R 3 with constant mean curvature -2 √ 3 .

We can now easily prove a correspondence (Theorem 5.3) between almost complex surfaces in S 3 × S 3 and harmonic maps in S 3 . The proof follows from Theorem 5.1 and the following proposition.

Proposition 5.2. To a harmonic map from a simply connected surface into the 3-sphere S 3 one can associate an H-surface X in Euclidean 3-space satisfying

(26) X xx + X yy = -2X x × X y ,
and vice versa. Moreover the harmonic map is non-conformal if and only if the associated H-surface has non-vanishing holomorphic differential X z , X z dz 2 where z = x + iy.

Proof. Consider a harmonic map f : S → S 3 and take a complex coordinate z = x + iy. Then f x = f α and f y = f β. From the integrability condition f xy = f yx we obtain

(27) α y -β x = 2α × β.
The map f is harmonic, so f xx + f yy is parallel with f . In terms of α and β this equation gives (28)

α x + β y = 0.
Since S is a simply connected surface, there exists a R 3 -valued map X, unique up to a real constant vector in R 3 , such that

X x = -β, X y = α.
Indeed, equation (28) exactly is the integrability condition for this system of differential equations. Equation (27) now becomes (26). If the harmonic map is nonconformal, we can assume that z is an adapted coordinate. Then clearly X x , X x -X y , X y = β, β -α, α = 4 is non-zero.

In order to prove the other implication, one proceeds through the previous construction in the opposite direction.

Theorem 5.3. To a harmonic map from a simply connected surface into the 3sphere S 3 one can associate an almost complex surface in S 3 × S 3 , and vice versa. Moreover the harmonic map is non-conformal if and only if the associated almost complex surface has a non-vanishing holomorphic differential. Proof. By Theorem 5.1 the harmonic map yields a H-surface X satisfying (26) (here H = -1). After a suitable dilation we can assume that the surface satisfies (25) (here H = -2 √ 3 ). Observe now that the quadratic differential of the Hsurface X vanishes if and only if the quadratic differential of the dilated surface vanishes. We then only have to apply Proposition 5.2 and the proof is done.

Remarks 5.4.

(1) Proposition 5.2 says that a harmonic map into the 3-sphere corresponds to an H-surface in Euclidean 3-space and vice versa. This is just the non-conformal analogue of the Lawson correspondence [START_REF] Lawson | Complete minimal surfaces in S 3[END_REF]: each minimal surface in S 3 has a constant mean curvature "cousin surface" in R 3 . By Lawson's correspondence Theorem almost complex surfaces with vanishing holomorphic differential in S 3 × S 3 are in correspondence with surfaces of constant mean curvature -2 √ 4 in R 3 and with minimal surfaces in S 3 .

(2) In [START_REF] Dioos | Flat almost complex surfaces in S 3 × S 3[END_REF] Li, Ma and the first and last author proved two Bonnet-type existence and uniqueness theorems for almost complex surfaces in S 3 ×S 3 . Combining Proposition 4.1 and Theorem 5.3 gives an alternative prove of this theorems.

Corollary 5.5. Let ψ : S → S 3 × S 3 be a simply connected almost complex surface with non-vanishing holomorphic differential. To this surface we can associate two almost complex surfaces that both have a non-vanishing holomorphic differential.

Proof. This is a simple consequence of Theorems 3.2 and 5.3. Let f be the nonconformal harmonic map corresponding to ψ via Theorem 5.3. Then the harmonic maps f + and f -obviously also correspond to two almost complex surfaces.

The remark after Theorem 3.5 tells us one can associate to an almost complex surface in S 3 × S 3 with non-vanishing holomorphic differential a sequence {ψ p | p ∈ Z} of such surfaces. One just has to define ψ p as the almost complex surface in S 3 × S 3 associated to the harmonic map f p for each p ∈ Z. In a similar fashion a H-surface in Euclidean 3-space induces a sequence {X p | p ∈ Z} of H-surfaces. The derivatives of X ε are X ε x = -β ε and X ε y = α ε . The expressions for α ε and β ε

Lemma 3 . 3 .

 33 The function φ ε is positive on an open dense subset U ε of S.

Theorem 4 . 3 .

 43 The following statements are equivalent. (a) a non-conformal harmonic map f is congruent to its ε-transform f ε ; (b) φ = φ ε and µ = µ ε ; (c) the function φ is constant; and (d) the function µ is constant and non-zero. If one of these statements holds, all maps in the sequence associated to f are the same up to congruence.Proof. The first equivalence is an immediate corollary of Proposition 4.1.

Belgian Interuniversity Attraction Pole P07/18 (Dygest).

Proof. Consider a map f of the form (18). An easy calculation gives

It follows from (19) that f x , f x = 4 sinh 2 φ, f x , f y = 0 and f y , f y = 4 cosh 2 φ, so f is non-conformal and the coordinate z = x + iy is adapted. The map f is harmonic if and only if

By the definition (20) of θ, this condition is satisfied. Now we only have to calculate µ = ∂f 1 , N . The normal N is N = s cos(ax + by), s sin(ax + by), -r cos(cx + dy), -r sin(cx + dy) .

Hence, the real and imaginary parts of µ are given respectively by

Since φ and µ are constant, f is congruent to its ε-transform by Theorem 4.3. The converse statement is now easy to prove. By the uniqueness clausule of Proposition 4.1 it suffices to show that for the maps (18) every positive real number φ and every complex number µ with modulus sinh 2φ can occur. For φ this is trivial. We will now show that every complex number µ 0 = sinh 2φ e iτ can occur. If we write r = cos ρ and s = sin ρ, the expression (21) for µ becomes Proof. Equation (a) follows immediately from φ = φ ε and equation ( 16). The compatibility conditions (6) for φ and φ ε give

given in Lemma 3.4 are complicated so it is not practical to integrate this system to obtain the associated H-surfaces X ε . An explicit expression for X ε in terms of the original H-surface X is given in the next lemma.

Lemma 5.6. Consider an H-surface X in the Euclidean 3-space with H = -1.

Let z = x + iy be an adapted coordinate and φ the function such that X x , X x = 4 cosh φ and X y , X y = 4 sinh φ. The ε-transform X ε of X is given by

and satisfies equation (24).

Proof. Recall that the definition of a H-surface is independent of the choice of complex coordinate, so we may assume that z is an adapted complex coordinate for X. By Theorem 5.3 it is sufficient to show that the derivatives of the right hand side of (29) are equal to -β ε and α ε respectively. The moving frame equations ( 5) in terms of α and β are A direct calculation using these equations shows that the derivatives of the expression in (29) are indeed -β ε and α ε .