Hans-Christoph Grunau

F R Éd Éric

Robert Dedicated

Patrizia Pucci

UNIFORM ESTIMATES FOR POLYHARMONIC GREEN FUNCTIONS IN DOMAINS WITH SMALL HOLES

Keywords: 35J40, 35B45 Polyharmonic Green function, domains with small holes, uniform estimates

The Green function G -∆,Ω for the Laplacian under Dirichlet boundary conditions in a bounded smooth domain Ω ⊂ R n enjoys in dimensions n ≥ 3 the estimate:

Here, en denotes the volume of the unit ball B = B 1 (0) ⊂ R n . This estimate follows from the maximum principle, the construction of G -∆,Ω and the explicit expression of a suitable fundamental solution.

When passing to the polyharmonic Green function G (-∆) k ,Ω under Dirichlet boundary conditions almost all forms of maximum or comparison principles fail: Green function estimates become an intricate subject and, according to works of Krasovskiȋ, multiplicative constants have to be used which heavily depend on the smoothness properties of the underlying domains.

In the present paper we study a singular family of domains by removing arbitrarily small holes from a fixed smooth domain in R n with n > 2k. We prove Green function estimates which are uniform even when the size of the hole approaches 0, i.e. when the curvature of the boundary becomes unbounded.

Introduction and main results

Given an arbitrary C 2k,θ -smooth bounded domain Ω ⊂ R n with exterior unit normal ν, n > 2k ≥ 2 and θ ∈ (0, 1), we define G Ω : Ω × Ω \ {(x, x) : x ∈ Ω} → R as the Green function of (-∆) k in the domain Ω with Dirichlet boundary condition. This means that for f ∈ C 0,θ (Ω) the unique solution u ∈ C 2k,θ (Ω) of the polyharmonic Dirichlet problem [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF] (-∆)

k u = f in Ω, u = ∂ ν u = . . . = ∂ (k-1) ν u = 0 on ∂Ω is given by u(x) = Ω
G Ω (x, y)f (y) dy.

In case of exterior domains, which will naturally arise in what follows, one has to add zero boundary conditions (i.e. decay to 0) at infinity in order to have the Green function well defined.

Date: November 15th 2012.

We are interested in pointwise estimates for G Ω . In the special case k = 1, i.e. the case of the usual Laplacian, these can be deduced by using the maximum principle. This yields that G Ω is positive und bounded from above by the fundamental solution, i.e. for n > 2 and any bounded smooth domain Ω ⊂ R n we have [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF] ∀x, y ∈ Ω, x = y, : 0 < G Ω (x, y) < 1 (n -2)ne n |x -y| 2-n .

Here, e n denotes the measure of the n-dimensional unit ball. One should observe that the constant in the right inequality is independent of Ω, even with respect to singular perturbations.

When passing to biharmonic or -more general-polyharmonic equations, i.e. the cases k ≥ 2, the maximum principle is no longer available and positivity issues remain valid only in a very weak and modified sense. Mathematical contributions to this topic go back at least to Boggio and Hadamard [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF][START_REF] Hadamard | Sur certains cas intéressants du problème biharmonique[END_REF]; these papers are also fundamental for subsequent works on estimating polyharmonic Green functions.

For an extensive discussion of related and more recent contributions one may see Grunau-Robert [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] and the monograph Gazzola-Grunau-Sweers [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF]. There is no obvious idea how to directly prove higher order analogues to estimate (2). However, employing the general Schauder and L p -theory developed by Agmon, Douglis, and Nirenberg [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF], Krasovskiȋ [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations[END_REF][START_REF]Isolation of the singularity of the Green's function[END_REF] proved that for any given bounded sufficiently smooth domain Ω ⊂ R n , n > 2k, there exists

C Ω > 0 such that (3) |G Ω (x, y)| ≤ C Ω |x -y| 2k-n for all x, y ∈ Ω, x = y.
The constant C Ω depends on C 2k,θ -properties of the boundary ∂Ω. In Krasovskiȋ's works, very general operators and boundary conditions were discussed. Applying these general results to our special polyharmonic Dirichlet problems originally required a higher degree of smoothness. However, it turns out that for our purposes, C 2k,θ -smoothness of ∂Ω suffices. For more detailed information on this issue we refer to Theorem 2 in the appendix. Estimate (3) can also be extended to the derivatives of Green functions: For any 0 ≤ r ≤ 2k, there exists C Ω,r such that (4) |∇ r y G Ω (x, y)| ≤ C Ω,r |x -y| 2k-n-r for all x, y ∈ Ω, x = y. Here, ∇ r y denotes any partial derivative with respect to y of order r. The constant C Ω in the Green function estimate (3) depends -as soon as k > 1heavily on the smoothness properties of ∂Ω. As long as one considers families of domains with uniform smoothness properties one may choose the same constant. In the present article, we exhibit families of domains with unbounded curvature, namely fixed domains Ω where we punch out arbitrarily small holes. For uniform Green function estimates, (3) can no longer be used since the curvature blows-up, and so does in general the constant C Ω . Nevertheless, we can prove the following uniform estimates.

Theorem 1.

Let Ω be a C 2k,θ -smooth bounded domain of R n with n > 2k, θ ∈ (0, 1), and k ≥ 1. We choose any point x 0 ∈ Ω. Let ω be a further C 2k,θ -smooth bounded domain of R n containing 0. We fix a number q ∈ (0, 1). Then there exists a constant C = C(Ω, ω, x 0 , q) > 0 such that that for all ε ∈ (0, q d(x0,∂Ω) diam(ω) ), we have that

|G Ωε (x, y)| ≤ C|x -y| 2k-n for all x, y ∈ Ω ε , x = y,
where Ω ε := Ω \ {x 0 + εω}. In particular, this estimate is uniform for ε 0.

Remark 1.

(i) In Proposition 3 we extensively discuss the invariance properties of the polyharmonic operator and the corresponding Green function under Möbius transforms of R n . As a consequence, for any such Möbius transform J of R n one has the same estimate for |G J(Ωε) | with the same constant as for |G Ωε | in Theorem 1 above. (ii) In small dimensions one observes a different behaviour of the Green function. Nakai and Sario [START_REF] Nakai | Green's function of the clamped punctured disk[END_REF] discussed the biharmonic case k = 2 in dimension n = 2 with the help of energy estimates and their approach can probably be used for any k ≥ 2 and any dimension n < 2k. In this small dimensions case some (in general not all) of the Dirichlet boundary conditions remain in x 0 even in the singular limit Ω 0 = Ω \ {x 0 }. This phenomenon cannot be expected in large dimensions n ≥ 2k.

It is then natural to ask whether in estimates like (4) we may also expect uniformity with respect to the family of domains (Ω ε ) ε . This, however, is not the case, even not for the Laplacian, i.e. k = 1. More precisely, we have the following:

Proposition 1.
Let Ω, q ∈ (0, 1), Ω ε , ε > 0, be as in Theorem 1. Then for all 1 ≤ r ≤ 2k, we have that

sup ε∈(0,qd(x0,∂Ω)/ diam(ω)) sup x,y∈Ωε, x =y |x -y| n-2k+r |∇ r y G Ωε (x, y)| = +∞.
As mentioned at the beginning, one has a comparison principle for (1) in general only in the second order case, i.e. if k = 1. In this case, G Ω > 0 holds true for any Ω, while if k ≥ 2 one has positivity G Ω > 0 only in very restricted classes of domains among which are balls (Boggio [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF]) and small perturbations of balls (Grunau-Robert [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF]). In general, however, one has sign change, i.e. G Ω ≥ 0. Already Hadamard [START_REF] Hadamard | Sur certains cas intéressants du problème biharmonique[END_REF] observed that this will occur in the biharmonic case in twodimensional annuli with very small inner radii, see also Nakai-Sario [START_REF] Nakai | Green's function of the clamped punctured disk[END_REF]. On the other hand, for fixed domains, the negative part will be "relatively" small. For more detailed information on this issue one may see Grunau-Robert [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF], Gazzola-Grunau-Robert [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF] and Grunau-Robert-Sweers [START_REF] Grunau | Optimal estimates from below for biharmonic Green functions[END_REF]. For instance, the authors proved in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] that for any C 4,θ -smooth bounded domain Ω ⊂ R n , n > 4, there exists

C Ω > 0 such that (G Ω ) -L ∞ (Ω×Ω) ≤ C Ω
, where G Ω is the Green function for (-∆) 2 with Dirichlet boundary condition. A natural question is to ask whether one may expect uniformity of this lower bound with respect to families of domains. As shown by the following proposition, the validity of this guess is equivalent to the nonnegativity of all Green functions: Proposition 2. We assume that n > 2k. The two following assertions are equivalent:

(i) There exists C(k, n, θ) depending only on k, n, θ such that such that

(G Ω ) -L ∞ (Ω×Ω) ≤ C(k, n, θ) for all C 2k,θ -smooth bounded domains Ω ⊂ R n . (ii) G Ω ≥ 0 for all C 2k,θ -smooth bounded domains Ω ⊂ R n .
Since (ii) is false for the higher order case k ≥ 2 (see the discussion and references in the monograph Gazzola-Grunau-Sweers [3, pp. 62/63 and 69/70]) we conclude that there is no uniform bound for negative parts of biharmonic and polyharmonic Green functions. We emphasise that we only discuss Dirichlet boundary conditions and that positivity issues may be quite different for other boundary conditions.

Notation: In the sequel, C(a, b, . . .) denotes a constant depending on ω, Ω, a, b, . . .. The same notation can be used for two different constants from line to line, and even in the same line.

Proofs

We start with proving Theorem 1 and proceed in several steps. In order to keep the exposition as simple as possible we shall prove the theorem for q = 1 42 . At the end of Section 2.3 we shall indicate how to modify the proof for larger q < 1. Without loss of generality, we assume that x 0 = 0 so that Ω ε := Ω \ εω.

2.1. The Green function in the exterior domain R n \ ω. Let ω be a C 2k,θ domain of R n such that 0 ∈ ω. We define

ω 0 := inv(R n \ ω) ∪ {0} , where inv : R n \ {0} → R n \ {0}, x → x |x| 2 .
We emphasise that the inversion inv is a special Möbius transform of R n and in particular conformal. The set ω 0 is a C 2k,θ -smooth bounded domain of R n containing 0. We define ( 5)

G (εω) c (x, y) := ε n-2k |y| 2k-n |x| 2k-n G ω0 (ε inv(x), ε inv(y))
for all x, y ∈ R n \ εω. The following proposition shows that this is indeed the polyharmonic Green function in (εω) c . In order to have this Green function well defined (and behaving well) we impose the condition on it to decay to 0 at inifinity.

Proposition 3. For any ϕ ∈ C 2k c (R n \ εω) such that ∂ (i)
ν ϕ = 0 on ∂(εω) for i = 0, . . . , (k -1), we have that

(6) ϕ(x) = R n \εω G (εω) c (x, y)(-∆) k ϕ(y) dy
for all x ∈ R n \ εω. Moreover, for all 0 ≤ i ≤ 2k, the derivatives with respect to y satisfy the upper bound

(7) |∇ i y G (εω) c (x, y)| ≤ C|y| -i r≤i |x| r |x -y| 2k-n-r .
We remark that the Green functions in exterior domains display a completely different behaviour in small dimensions n ≤ 2k. In this case there are no bounds in (εω) c which are uniform with respect to ε 0.

Proof. We prove the claim first for

ε = 1. Let ϕ ∈ C 2k c (R n \ ω) be such that ∂ (i)
ν ϕ = 0 on ∂ω for i = 0, . . . , (k -1). We show that [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations[END_REF] ϕ

(x) = R n \ω |y| 2k-n |x| 2k-n G ω0 (inv(x), inv(y))(-∆) k ϕ(y) dy for all x ∈ R n \ ω.
Indeed, inv is the composition of two sterographic projections of opposite poles, and therefore, it is conformal and the pull-back of the Euclidean metric Eucl via inv is inv Eucl = | • | -4 Eucl = µ 4/(n-2k) Eucl where µ(x) := |x| 2k-n for all x ∈ R n \ {0}.

As a consequence, considering (-∆) k as the conformal operator of Graham-Jenne-Mason-Sparling for the Euclidean space (see [START_REF] Graham | Conformally invariant powers of the Laplacian. I. Existence[END_REF]), the conformal law of the GJMS operators yields

((-∆) k ϕ) • inv = µ -(n+2k)/(n-2k) (-∆) k (µ(ϕ • inv)).
In addition, the Jacobian of inv and then the Riemannian element of volume of inv Eucl are Jac(inv

) = | • | -2n and dv inv Eucl = | • | -2n dx.
This transformation behaviour of polyharmonic operators with respect to Möbius transforms is classical, see e.g. Loewner [START_REF] Ch | On generation of solutions of the biharmonic equation in the plane by conformal mappings[END_REF] and references therein. A convenient and easily accessible reference is also Gazzola-Grunau-Sweers [3, Lemma 6.14] .

We fix x ∈ R n \ω and we consider x := inv(x) ∈ ω 0 \{0}. We define φ(y) := µ(y)ϕ• inv(y) = |y| 2k-n ϕ(y/|y| 2 ) for y ∈ ω 0 \{0}. We find that φ is vanishing around 0 and therefore extends smoothly to ω 0 . It follows from Green's representation formula that

φ(x ) = ω0 G ω0 (x , y)(-∆) k φ(y) dy.
Performing the change of variable y = inv(z) and using the above properties yields

φ(x ) = R n \ω |z| n+2k G ω0 (x , inv(z))(-∆) k ϕ(z)|z| -2n dz.
Going back to the expression of ϕ yields [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations[END_REF].

Given α a multi-index and j ∈ {1, . . . , n}, there exists an homogeneous polynomial P α j of degree |α| + 1 such that ∂ α inv(x) j = P α j (x) |x| 2(|α|+1) for all x ∈ R n \ {0}, where |α| is the length of the index. We fix x, y ∈ R n \ ω such that x = y. With help of the binomial formula, the derivative of order α with respect to y is such that

(9) |∂ α y G ω c (x, y)| ≤ C|x| 2k-n β≤α |y| 2k-n-|α|+|β| |∂ β y (G ω0 (inv(x), inv(y)))|,
where we have adopted the standard order on multi-indices. For |β| ≥ 1, the chain rule yields It follows from Krasovskiȋ [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations[END_REF][START_REF]Isolation of the singularity of the Green's function[END_REF] that for any 0 ≤ r ≤ 2k, there exists

∂ β (f • inv) = 1≤r≤|β|
C = C(ω 0 = inv(ω c ) ∪ {0}, r) > 0 such that |∇ r G ω0 (x, y)| ≤ C|x -y| 2k-n-r
for all x, y ∈ ω 0 , x = y. For the sake of completeness, we refer to Theorem 2 in the appendix where we comment on an alternative to Krasovskiȋ's proof. Noting that

(11) | inv(x) -inv(y)| = |x -y| |x| • |y|
and putting ( 9), ( 10) and ( 11) together yields

|∂ α y G ω c (x, y)| ≤ C|y| -|α| r≤|α| |x| r |x -y| 2k-n-r .
This proves the claim for ε = 1, while for arbitrary ε > 0 it follows from the previous reasoning and the observation that

G (εω) c (x, y) := ε 2k-n G ω c (x/ε, y/ε).
2.2. Control outside a small annulus. Given δ ∈ 0, d(0,∂Ω)

3 , we define η δ ∈ C ∞ c (Ω) such that η δ (x) = 1 for all x ∈ B δ (0) and η δ (x) = 0 for all x ∈ Ω \ B 2δ (0). Given ε ∈ (0, δ 2 diam(ω) ) and x, y ∈ Ω ε , we define (12) Gε,δ (x, y) := η δ (y)G (εω) c (x, y) + (1 -η δ (y))G Ω (x, y).
We get that

(-∆) k Gε,δ (x, • ) = η δ (-∆) k G (εω) c (x, • ) + (1 -η δ )(-∆) k G Ω (x, • ) + i<2k A i (∇ 2k-i η δ , ∇ i G (εω) c (x, • )) + A i (∇ 2k-i (1 -η δ ), ∇ i G Ω (x, • )) ,
where the A i s are contractions of suitable tensors, that is bilinear forms with smooth coefficients. Therefore, for any x ∈ Ω ε , there exists f ε,δ,x such that

(-∆) k Gε,δ (x, • ) = δ x + f ε,δ,x in D (Ω ε ).
Moreover, the pointwise control (7) yields

|f ε,δ,x (y)| ≤ C • 1 B 2δ (0)\B δ (0) |x -y| 1-n
for all x, y ∈ Ω ε , x = y. In particular, there exists C(δ) > 0 such that

(13) f ε,δ,x L ∞ (Ωε) ≤ C(δ) for all ε > 0 and x ∈ Ω ε,δ
where

Ω ε,δ := Ω ε ∩ B δ/2 (0) ∪ Ω ε \ B 3δ (0) = Ω ε \ B 3δ (0) \ B δ/2 (0) .
Then it follows from elliptic theory that for any x ∈ Ω ε,δ , there exists

u x,ε,δ ∈ W k,2 0 (Ω ε ) such that (14) (-∆) k u x,ε,δ = f ε,δ,x in Ω ε , ∂ (i)
ν u x,ε,δ = 0 for all i = 0, . . . , k -1 on ∂Ω ε .

We claim that u x,ε,δ ∈ C 2k-1 (Ω ε ) for all ε, δ > 0 and x ∈ Ω ε,δ . Moreover, there exists C(δ) > 0 such that (15)

u x,ε,δ C 2k-1 (Ω\B δ/4 (0)) ≤ C(δ)
for all admissible ε, δ > 0 and x ∈ Ω ε,δ .

We prove this claim. For simplicity, we define

((-∆) k/2 ψ) 2 := ((-∆) l ψ) 2 if k = 2l is even |∇(-∆) l ψ| 2 if k = 2l + 1 is odd.
As a consequence, u → (-∆) k/2 u 2 is a norm on W k,2 0 (Ω ε ), the completion of C ∞ c (Ω ε ) for the usual norm. Multiplying (14) by u x,ε,δ and integrating by parts yields with Hölder's inequality

Ω ((-∆) k/2 u x,ε,δ ) 2 dx = Ωε ((-∆) k/2 u x,ε,δ ) 2 dx = Ωε f ε,δ,x u x,ε,δ dy ≤ f ε,δ,x 2n n+2k u x,ε,δ 2n n-2k . Sobolev's inequality yields the existence of C n,k > 0 such that u 2n n-2k ≤ C n,k (-∆) k/2 u 2 for all u ∈ C ∞ c (R n ). The density of C ∞ c (Ω ε ) in W k,2 0 (Ω ε ) allows to conclude that u x,ε,δ 2 2n n-2k ≤ C 2 n,k f ε,δ,x 2n n+2k u x,ε,δ 2n n-2k
for all ε > 0 and x ∈ Ω ε,δ . Therefore u x,ε,δ 2n n-2k ≤ C (δ). It follows from elliptic theory (see for instance Agmon-Douglis-Nirenberg [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF]) that for all p > 1 and all δ > 0, there exists C(δ ) > 0 such that

u x,ε,δ W 2k,p (Ω\B δ (0)) ≤ C(δ , p, Ω)( f ε,δ,x p + u x,ε,δ p ).
The claim (15) follows from this inequality, Sobolev's inequalities and iterations.

It remains to gain control of u x,ε,δ in B δ (0) \ (εω). To this end we consider η δ u x,ε,δ and observe that this function solves a Dirichlet problem in the exterior domain (εω) c . Indeed, we have that

(-∆) k (η δ u x,ε,δ ) = η δ (-∆) k u x,ε,δ + i<2k A i (∇ 2k-i η δ , ∇ i u x,ε,δ ) = η δ f ε,δ,x + i<2k A i (∇ 2k-i η δ , ∇ i u x,ε,δ ) := fε,δ,x
where the A i 's are as above. We observe that supp fε,δ,x ⊂ B 2δ (0) and fε,δ,x ∞ ≤ C(δ) for all x ∈ Ω ε,δ . Since η δ u x,ε,δ has compact support in R n \ εω and vanishes up to (k -1) th order on ∂(εω), Green's representation formula (6) yields

(η δ u x,ε,δ )(z) = R n \εω G (εω) c (z, y) fε,δ,x (y) dy for all z ∈ R n \ εω. Consequently, for any z ∈ B δ (0) \ (εω), one gets |u x,ε,δ (z)| = |(η δ u x,ε,δ )(z)| ≤ R n \εω |G (εω) c (z, y) fε,δ,x (y)| dy ≤ C(δ) B 2δ(0) \εω |y -z| 2k-n dz ≤ C(δ).
This inequality combined with (15) yields ( 16)

u x,ε,δ L ∞ (Ωε) ≤ C(δ) for all x ∈ Ω ε,δ .
As a consequence, we find that

(-∆) k ( Gε,δ (x, • ) -u x,ε,δ ) = δ x weakly in D (Ω ε )
and

∂ (i)
ν ( Gε,δ (x, • ) -u x,ε,δ ) = 0 on ∂Ω ε for all x ∈ Ω ε,δ and all i = 0, . . . , k -1. The uniqueness of the Green function implies that

(17) G Ωε (x, • ) = Gε,δ (x, • ) -u x,ε,δ
and then, using ( 6) and ( 16), we arrive at

|G Ωε (x, y)| ≤ | Gε,δ (x, y)| + |u x,ε,δ (y)| ≤ C(Ω, ω)|x -y| 2k-n + C(δ) (18) ≤ C(δ)|x -y| 2k-n (19) for all x ∈ Ω ε,δ = (Ω ε ∩ B δ/2 ) ∪ (Ω ε \ B 3δ (0)
) and all y ∈ Ω ε .

Conclusion of the proof of Theorem 1.

We fix δ 0 ∈ (0, d(x0,∂Ω)

21

). We apply Section 2.2 with δ := δ 0 and to δ := 7δ 0 . Since Ω ε = Ω ε,δ0 ∪ Ω ε,3δ0 , it follows from (18) that there exists C > 0 such that

|G Ω\εω (x, y)| ≤ C|x -y| 2k-n for all x, y ∈ Ω \ εω, x = y.
This proves Theorem 1 for q = 1/42. For q ∈ (1/42, 1), instead of δ/2, δ, 2δ, 3δ, in Section 2.2 one has to work with δ/(1 + σ), δ, (1 + σ)δ, (1 + 2σ)δ with σ > 0 sufficiently close to 0. Alternatively one may argue that for ε ∈ [(1/42)d(x 0 , ∂Ω)/ diam(ω), qd(x 0 , ∂Ω)/ diam(ω)) the boundaries of the Ω ε enjoy uniform C 2k,θ -properties so that (3) holds uniformly with respect to these ε.

  I1+...+Ir=β j1,...,jr c (I1,...,Ir) j1,...,jr ∂ I1 inv j1 . . . ∂ Ir inv jr (∂ j1...jr f ) • inv for any function f when the derivatives make sense. The second sum is taken over all decompositions of β as a sum of r multi-indices and the c (I1,...,Ir) j1,...,jr are combinatorial constants which can be calculated explicitly. When restricting to suitable decompositions of β these constants are equal to 1. This formula yields (10) |∂ β y (G ω0 (inv(x), inv(y)))| ≤ C r≤|β| |y| -|β|-r |(∇ r G ω0 )(inv(x), inv(y))| for all β ≤ α. Here, ∇ r f = (∂ γ f ) |γ|=r when this makes sense.

2. 4 .

 4 Proof of Proposition 1. We argue by contradiction and assume that there exist 1 ≤ r ≤ 2k and C > 0 such that (20)|x -y| n-2k+r |∇ r y G Ωε (x, y)| ≤ C for all x, y ∈ Ω ε , x = y, uniformly in ε → 0. For any x, y ∈ (ε -1 Ω) \ ω, we define G ε (x, y) := ε n-2k G Ωε (εx, εy). It follows from[START_REF] Nicolesco | Les fonctions polyharmoniques[END_REF], (15), and (17) that for any x ∈ ω c , we have that limε→0 G ε (x, y) = G ω c (x, y) in C 0 loc (R n \ (ω ∪ {x})). Since (-∆) k G ε (x, • ) = 0 and G ε (x,• ) vanishes on ∂ω up to order (k -1), elliptic regularity yields convergence in C 2k loc (ω c \ {x}). Rewriting (20) for G ε and passing to the limit ε → 0 yields(21) |x -y| n-2k+r |∇ r y G ω c (x, y)| ≤ C for all x, y ∈ R n \ ω, x = y. We fix x = 0 and we define G R (z) := R n-2k G ω c (Rx, z) for all z ∈ ω c and R > R 0 large enough. It follows from the explicit expression of G ω c in (5) that (22) lim R→+∞ G R (z) = G(z) := |x| 2k-n |z| 2k-n G ω0 0, z |z| 2 in C 0 loc (R n \ ω). Since (-∆) k G R = 0and G R vanishes on ∂ω up to order (k -1), elliptic regularity yields the convergence of (G R ) to G in C 2k loc (ω c ). On the other hand, (21) may be rewritten as|∇ r G R (z)| ≤ CR -r |x -R -1 z| 2k-n-r

Acknowledgement. We are grateful to the referee for helpful remarks on a previous version of this paper.

for z in a compact sudomain of R n \ ω and R large enough. Since r ≥ 1, passing to the limit R → +∞ yields ∇ r G = 0 in R n \ ω, which contradicts the explicit expression (22) of G. This concludes the proof of Proposition 1.

2.5. Proof of Proposition 2. Assume that (ii) does not hold. Then there exists a C 2k,θ -smooth bouned domain ω 0 ⊂ R n such that G ω0 attains some negative values, say at (x 0 , y 0 ) ∈ ω 0 ×ω 0 , x = 0. We define ω := (inv(R n \ω 0 ))∪{0} and Ω ε := Ω\εω where ε > 0 is small and Ω is a smooth bounded domain containing 0. It follows from ( 12), (16), and (17) that

and then (i) does not hold. Conversely, if (ii) holds, then (i) holds.

Appendix A. Pointwise control of the Green function for fixed domains

The following result, under stronger smoothness assumptions on Ω but at the same time in a more general context, is due to Krasovskiȋ [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations[END_REF][START_REF]Isolation of the singularity of the Green's function[END_REF]: We sketch here an alternative proof.

Proof. The case r = 0 and k = 2 under the smoothness assumptions as in the theorem is treated in Grunau-Robert [5, Theorem 4] (see also Gazzola-Grunau-Sweers [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF]Propositions 4.22 and 4.23] for an exposition in book form). By making the obvious changes one may check that the proof can be extended to any k ≥ 1 and n > 2k. (Only the discussion of the smaller dimensions n ≤ 2k requires more care.) This means that there exists a constant C(Ω) > 0 such that (24)

for all x, y ∈ Ω, x = y. We fix r ≥ 1 and we prove (23) by using local elliptic estimates and rescaling arguments. We proceed as in Grunau-Gazzola-Sweers [3, Prop.

The constant is uniform in R; the behaviour with respect to (small) R is obtained by means of scaling.

Keeping x ∈ Ω fixed, for any y ∈ Ω \ {x} we choose R = |x -y|/4 and apply (25) and (24) in B R (y) ⊂ B 2R (y) to G Ω (x, • ). This proves (23).


Hans-Christoph Grunau
email: hans-christoph.grunau@ovgu.de

F R Éd

AND Éric Robert
email: frederic.robert@iecn.u-nancy.fr

Guido Sweers
email: gsweers@math.uni-koeln.de

OPTIMAL ESTIMATES FROM BELOW FOR BIHARMONIC GREEN FUNCTIONS

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary C 4,γ -smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.

Introduction

It is well-known that the Green function G(x, y) for second order elliptic equations on bounded domains can be estimated from above and from below by positive multiples of the same (positive) function, which is explicitly given in terms of the distances to the boundary, d(x), d(y), and the distance |x -y|. See for example [START_REF] Lai | From Brownian motion to Schrödinger's equation[END_REF]. The behaviour of the biharmonic Green function for Dirichlet boundary conditions should be somehow similar but will have two crucial distinctions. The singularity of course is of lower order, namely n -4 instead of n -2 with n the dimension, but a more serious distinction is the fact that the biharmonic Green function is not everywhere positive for most domains. Indeed, the few known domains with a biharmonic Green function of a fixed positive sign are balls in arbitrary dimensions, small perturbations of those balls and of some limaçons in 2 dimensions. See respectively [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF][START_REF] Grunau | Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions[END_REF][START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] and [START_REF] Dall'acqua | On domains for which the clamped plate system is positivity preserving[END_REF][START_REF] Dall'acqua | The clamped-plate equation for the limaçon[END_REF]. The results in [START_REF] Dall'acqua | The clamped-plate equation for the limaçon[END_REF] extend and correct [START_REF] Hadamard | Sur certains cas intéressants du problème biharmonique[END_REF].

It has been observed numerically on domains with a sign changing biharmonic Green function that the negative part is rather small and that it is also not located near the singularity. The aim of this paper is to give optimal estimates from below. Previous results concerning smallness of the negative part have been obtained in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] for n ≥ 3 and [START_REF] Dall'acqua | Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains[END_REF] for n = 2. With the estimates for the absolute value of that Green function in [START_REF] Dall'acqua | Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems[END_REF] these previously known estimates are, when n > 4, as follows: and G Ω denotes the said Green function. Let us remind the reader that the biharmonic Green function G Ω is such that

u(x) = Ω G Ω (x, y) f (y) dy solves (2) ∆ 2 u = f in Ω, u = |∇u| = 0 on ∂Ω.
In (1) the dimension is restricted to n > 4. As has been shown in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] for n = 3, 4 and in [START_REF] Dall'acqua | Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains[END_REF] for n = 2, the estimate from below in [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I[END_REF] holds in all dimensions. An estimate from above has also been proved for n ≤ 4 but the formula for that estimate is different from [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I[END_REF]. Those estimates can be found in [START_REF] Dall'acqua | Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems[END_REF] and contain the function H Ω in [START_REF] Dall'acqua | Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains[END_REF].

The main result is the estimate from below in the following theorem. For the sake of completeness we include the estimate from above. [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF]. Then there exist constants c 1 ≥ 0, c 2 > 0, depending on the domain Ω, such that we have the following Green function estimate:

Theorem 1. Let Ω ⊂ R n (n ≥ 2) be a bounded C 4,γ -smooth domain. Let G Ω denote the biharmonic Green function in Ω for
(3) c -1 2 H Ω (x, y) ≤ G Ω (x, y) + c 1 d(x) 2 d(y) 2 ≤ c 2 H Ω (x, y) for all x, y ∈ Ω, where (4)
H Ω (x, y) :=

                         |x -y| 4-n min 1, d(x) 2 d(y) 2 |x -y| 4 if n > 4, log 1 + d(x) 2 d(y) 2 |x -y| 4 if n = 4, d(x) 2-n/2 d(y) 2-n/2 min 1, d(x) n/2 d(y) n/2 |x -y| n if n = 2, 3.
x G x,y y n 4

Figure 1. The dashed curve gives the typical behaviour of G Ω ; the shaded area describes the band given by (3).

The estimate from above follows from [START_REF] Dall'acqua | Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems[END_REF][START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations (Russian)[END_REF][START_REF] Ju | Isolation of singularities of the Green's function (Russian)[END_REF] so that only the estimate from below has to be proved here. One should observe that G Ω (x, y) + c 1 d(x) 2 d(y) 2 ≥ 0 was proved for some suitable c 1 ≥ 0 in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF]. The preceding result may be considered as an extension of the estimates in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] showing that close to the pole the positive singular behaviour of the fundamental solution can also be seen in the Green function.

Due to the different behaviour of the Green function we need to distinguish between n ≥ 3 and n = 2 in proving Theorem 1. Finally we should remark that the lack of a maximum principle not only results in sign changing Green functions but also complicates the proof of these estimates in the fourth order case. The proof in the second order case heavily depends on the Harnack inequality which in turn depends on the maximum principle.

An interesting consequence of Theorem 1 is a uniform local positivity result. When n ≥ 3 this was proved in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF], while for n = 2 we refer to [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF]Theorem 6.15]. Here the emphasis is on the interplay between Theorem 1 and the following result. Moreover, we provide a proof for n = 2 which is much simpler than the previous one and in the same spirit as for n ≥ 3.

Theorem 2. Let Ω ⊂ R n (n ≥ 2) be a bounded C 4,γ -smooth domain. Let G Ω denote the biharmonic Green function in Ω for [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF]. Then there exist a constant r Ω > 0, such that [START_REF] Dall'acqua | Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems[END_REF] G Ω (x, y) > 0 for all x, y ∈ Ω with |x -y| < r Ω .

Some auxiliary results for n ≥ 3

A careful inspection of the proofs in Nehari [START_REF] Nehari | On the biharmonic Green's function[END_REF] and Grunau-Sweers [START_REF] Grunau | Regions of positivity for polyharmonic Green functions in arbitrary domains[END_REF] shows the following local estimate for the biharmonic Green function from below. Proposition 3. Let n ≥ 3. Then there exists constants δ n > 0 and c 3 > 0, which depend only on the dimension n, such that the following holds true.

Assume Ω ⊂ R n to be a C 4,γ -smooth bounded domain and let G Ω := G ∆ 2 ,Ω denote the Green function for the biharmonic operator under Dirichlet boundary conditions. If [START_REF] Dall'acqua | On domains for which the clamped plate system is positivity preserving[END_REF] |x -y| ≤ δ n max{d(x), d(y)}, then we have

G Ω (x, y) >              c 3 |x -y| 4-n if n > 4, c 3 log 1 + 1 |x -y| 4 if n = 4, c 3 d(x) 1/2 d(y) 1/2 if n = 3.
For the constant δ n , one may achieve that δ n ≥ 0.5.

In dimension n = 2 it seems impossible to achieve a global linear dependence of the radius of a ball of guaranteed positivity on the boundary distance, see Lemma 7 and [17]. In other words, the best result seems δ 2 = δ 2 (Ω), which strictly depends on Ω.

Due to Proposition 3 and using the same constants δ n > 0 as there we may restrict ourselves in what follows to x, y such that [START_REF] Dall'acqua | The clamped-plate equation for the limaçon[END_REF] x, y ∈ Ω, x = y, |x -y| > δ n max{d(x), d(y)}.

Lemma 4. Suppose that n ≥ 3 and that Ω ⊂ R n is a bounded C 4,γ -smooth domain.

Then for each x 0 ∈ Ω there exists a radius r = r x0 > 0 and a constant C = C x0 > 0 such that for all x, y ∈ Ω x0,r := Ω ∩ B r (x 0 ) subject to condition [START_REF] Dall'acqua | The clamped-plate equation for the limaçon[END_REF] one has

(8) G Ω (x, y) ≥ C|x -y| -n d(x) 2 d(y) 2 .
Proof. We only need to discuss x 0 ∈ ∂Ω since for interior points x 0 one may choose r = r x0 > 0 so small that condition [START_REF] Dall'acqua | The clamped-plate equation for the limaçon[END_REF] becomes void. We assume by contradiction that there exist

x k , y k ∈ Ω x0,1/k = Ω ∩ B 1/k (x 0 ) subject to (7) such that (9) G Ω (x k , y k ) < 1 k |x k -y k | -n d(x k ) 2 d(y k ) 2 .
In particular we have

x k → x 0 , y k → x 0 , d(x k ), d(y k ) → 0, |x k -y k | → 0.
Without loss of generality we may assume that x 0 = 0 and that the first unit vector e 1 is the exterior unit normal to ∂Ω at x 0 .

Ω

e 1 x 0 Ω x0,1/k Figure 2
. Ω and subdomain Ω x0,1/k for x 0 ∈ ∂Ω.

For k large enough, we may define xk ∈ ∂Ω as the closest boundary point to x k . We introduce the rescaled biharmonic Green functions

G k (ξ, η) := |x k -y k | n-4 G Ω (x k + |x k -y k |ξ, xk + |x k -y k |η) for ξ, η ∈ Ω k := 1 |x k -y k | (-x k + Ω) .
Since xk → 0, the exterior unit normal at ∂Ω converges to the first unit vector and so we conclude that

Ω k → H := {x : x 1 < 0} locally uniformly for k → ∞.
It was proved in [10, Lemma 7] that locally uniformly in

H × H \ {(ξ, ξ); ξ ∈ H} G k (ξ, η) → G H (ξ, η) = 1 4ne n |ξ -η| 4-n |ξ * -η|/|ξ-η| 1 (v 2 -1)v 1-n dv,
where ξ * = (-ξ 1 , ξ 2 , . . . , ξ n ) and e n is the n-dimensional volume of B 1 (0) ⊂ R n . We remark that this step and in particular the required uniqueness proof for G H were carried out in Grunau-Robert [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] using the assumption n ≥ 3. The necessary modifications for n = 2 are emphasized in the proofs of Lemmata 7 and 6. Assumption [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF] gives

(10) G k (ξ k , η k ) = |x k -y k | n-4 G Ω (x k , y k ) < 1 k |x k -y k | -4 d(x k ) 2 d(y k ) 2 ,
where

ξ k = 1 |x k -y k | (x k -xk ), η k = 1 |x k -y k | (y k -xk ), |ξ k | = d(x k ) |x k -y k | ≤ 1 δ n , |ξ k -η k | = 1.
After passing to a further subsequence we find ξ, η ∈ H with ξ = lim k→∞ ξ k , η = lim k→∞ η k . In view of the local smooth convergence of G k to the biharmonic Green function G H in the half space H, Boggio's formula and |ξ -η| = 1 there exists a positive constant σ > 0 such that for k large enough:

G k (ξ k , η k ) ≥ σd(ξ k ) 2 d(η k ) 2 = σ d(x k ) |x k -y k | 2 d(y k ) |x k -y k | 2 = σ|x k -y k | -4 d(x k ) 2 d(y k ) 2 .
This contradicts [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] and the proof of the lemma is complete.

3. Proof of the main estimate for n ≥ 3 Supposing that (6) holds, i.e. |x -y| ≤ δ n max{d(x), d(y)}, one finds that even (1 -δ n )|x -y| ≤ δ n min{d(x), d(y)} and hence

d(x)d(y) |x -y| 2 ≥ 1 -δ n δ 2 n .
So in that case the estimate from below in Theorem 1 follows directly from Proposition 3. Hence, we may assume from now on, again, that x, y are subject to condition [START_REF] Dall'acqua | The clamped-plate equation for the limaçon[END_REF]. Applying a compactness argument to

Ω = x0∈Ω Ω x0,rx 0 /2
we see that there exist positive numbers r > 0,

c 4 > 0, such that |x -y| ≤ r implies that G Ω (x, y) ≥ c 4 |x -y| -n d(x) 2 d(y) 2 .
If |x -y| ≥ r we take from [START_REF] Dall'acqua | Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems[END_REF], cf. also [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF], that G Ω (x, y) ≥ -c 5 |x -y| -n d(x) 2 d(y) 2 so that

G Ω (x, y) + 2c 5 |x -y| -n d(x) 2 d(y) 2 ≥ c 5 |x -y| -n d(x) 2 d(y) 2 .
Since |x -y| -n ≤ r -n we end up with

G Ω (x, y) + c 6 d(x) 2 d(y) 2 ≥ c 5 |x -y| -n d(x) 2 d(y) 2
and positive constants c 5 , c 6 > 0 in this case. The proof of Theorem 1 for n ≥ 3 is complete.

Auxiliary results for n = 2

Lemma 5. Let H = {x ∈ R 2 : x 1 < 0} and let G ∈ C 4 (H × H \ {(x, x) : x ∈ H}) be a biharmonic Green function with Dirichlet boundary condition, that is

H G(x, . )∆ 2 ϕ dy = ϕ(x) + ∂H ∆ G(x, . )∂ ν ϕ -ϕ∂ ν ∆ G(x, .
) dσ for all ϕ ∈ C 4 c (H) and x ∈ Ω. Moreover, we assume that G(x, y) = G(y, x) for all x = y and that a growth condition holds at infinity:

(11) | G(x, y)| ≤ C 1 + |x| 2 + |y| 2 1 + (log |x|) + + (log |y|) + .
Then G is uniquely determined and given by Boggio's formula [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF]:

G(x, y) = G H (x, y) = 1 8π |x -y| 2 |x * -y|/|x-y| 1 v 2 -1 v dv.
with x * = (-x 1 , x 2 ).

Proof. We choose some arbitrary x ∈ H and keep it fixed in what follows. We write

G(x, y) = G H (x, y) + H(x, y),
where H is a regular function in H × H and ∆ 2 y H(x, . ) ≡ 0 in H, H(x, y) = ∂ y1 H(x, y) = 0 for y 1 = 0. According to [START_REF] Duffin | Continuation of biharmonic functions by reflection[END_REF][START_REF] Huber | On the reflection principle for polyharmonic functions[END_REF], or by checking directly, with y * = (-y 1 , y 2 ),

H * (x, y) :=    H(x, y) if y 1 ≤ 0, -H(x, y * ) -2y 1 ∂H ∂y 1 (x, y * ) -y 2 1 (∆ y H) (x, y * ) if y 1 > 0,
satisfies H * (x, . ) ∈ C 4 R 2 and is biharmonic on R 2 . Since G H (x, y) satisfies [START_REF] Grunau | Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions[END_REF], so does H(x, y). So we have

|H(x, y)| ≤ C x 1 + |y| 2 (log |y|) + .
Using local elliptic estimates and their scaling behaviour for biharmonic functions satisfying Dirichlet boundary conditions on ∂H, that is,

D α H(x, . ) L ∞ (B R ∩H) ≤ C R |α| H(x, . ) L ∞ (B 2R ∩H) we find that |∇ y H(x, y)| ≤ C x 1 + |y| (log |y|) + , |∇ 2 y H(x, y)| ≤ C x 1 + (log |y|) + .
Having these estimates for H(x, y) and these first two derivatives, we have an estimate for H * (x, y) and may repeat the above arguments to find similar estimates for the derivatives of H * (x, y) and even

|∇ 3 y H * (x, y)| ≤ C x 1 + (log |y|) + 1 + |y| .
The maximum principle applied to the harmonic function ∇ y ∆ y H * (x, . ) shows that

∇ y ∆ y H * (x, . ) C 0 (B R (0)) ≤ C (1 + |log |R||) 1 + |R| . Letting R → ∞ yields ∇ y ∆ y H * (x, . ) ≡ 0, ∆ y H * (x, . ) = a(x)
with a suitable function a(. ). This shows that any ∇ 3 y H * (x, . ) is harmonic and, as shown above, ∇ 3 y H * (x, y) → 0 as y → ∞. Hence, any ∇ 3 y H * (x, . ) ≡ 0 and by Taylor's formula and observing the boundary data we conclude that

H * (x, y) = b(x)y 2 1
with a suitable function b(. ). By symmetry H(x, y) = H(y, x) and so

H(x, y) = b(y)x 2 1 = b(x)y 2 1 = bx 2 1 y 2 1
, where b ∈ R is a suitable constant. Finally, the growth condition leads to b = 0, H(x, . ) ≡ 0, and G = G H . Lemma 6 (Estimates near the boundary). Suppose that n = 2 and that Ω ⊂ R 2 is a bounded C 4,γ -smooth domain. Then for each x 0 ∈ ∂Ω there exists a radius r = r x0 > 0 and a constant C = C x0 > 0 such that for all x, y ∈ Ω x0,r := Ω∩B r (x 0 ) one has

(12) G Ω (x, y) ≥ Cd(x)d(y) min 1, d(x)d(y) |x -y| 2 .
Proof. We assume by contradiction that there exist

x k , y k ∈ Ω x0,1/k = Ω∩B 1/k (x 0 ) such that (13) G Ω (x k , y k ) < 1 k d(x k )d(y k ) min 1, d(x k )d(y k ) |x k -y k | 2 .
In particular we have

x k → x 0 , y k → x 0 , d(x k ), d(y k ) → 0, |x k -y k | → 0.
Without loss of generality we may assume that x 0 = 0 and that the first unit vector e 1 is the exterior unit normal to ∂Ω at x 0 . After possibly passing to a subsequence it is enough to consider one of the following two cases.

First case: |x k -y k | ≥ 1 2 max{d(x k ), d(y k )}. This proof is as above for Lemma 4; only proving the required uniform bounds for the G k is slightly more involved. The arguments are sketched below in the second case. Thanks to Lemma 5 the convergence proof of [10, Lemma 7] can be extended to n = 2. One should observe that also the symmetry carries over to the limit.

Second case: |x

k -y k | < 1 2 max{d(x k ), d(y k )}. Observe that in this case d(x k ) < 2d(y k ) < 4d(x k ) and |x k -y k | < min{d(x k ), d(y k )}.
The assumption gives that

(14) G Ω (x k , y k ) < 1 k d(x k )d(y k )
In this case we rescale differently, however, xk ∈ ∂Ω denotes again the closest boundary point to x k . We introduce the rescaled biharmonic Green functions

G k (ξ, η) := d(x k ) -2 G Ω (x k + d(x k )ξ, xk + d(x k )η) for ξ, η ∈ Ω k := 1 d(x k ) (-x k + Ω) .
Since xk → 0, the exterior unit normal at ∂Ω converges to the first unit vector and so we conclude that Ω k → H := {x : x 1 < 0} locally uniformly for k → ∞.

For

ξ k = 1 d(x k ) (x k -xk ), η k = 1 d(x k ) (y k -xk )
the assumption ( 14) transforms into

(15) G k (ξ k , η k ) < 1 k d k (ξ k )d k (η k ) < 2 k ,
where d k := d( . , ∂Ω k ). Since ξ k , η k are bounded and their boundary distances are uniformly bounded from below by 1/2 we find after passing to a further subsequence that

ξ k → ξ ∞ ∈ H, η k → η ∞ ∈ H.
We claim that we have local uniform convergence in H × H (including the diagonal) of G k to G H , since we are in dimension n = 2.

To see this we observe first that Krasovskiȋ-type estimates (see [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations (Russian)[END_REF][START_REF] Ju | Isolation of singularities of the Green's function (Russian)[END_REF] and also [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF]Theorem 4.20]) yield at a first instance useful information only for the third derivatives. We have

|∇ 3 ξ,η G k (ξ, η)| ≤ C |ξ -η| uniformly in k.
Making use of Now, one may proceed further as in [10, Lemma 7]. So, [START_REF] Ju | Investigation of potentials connected with boundary value problems for elliptic equations (Russian)[END_REF] yields G H (ξ ∞ , η ∞ ) ≤ 0, while Boggio's formula shows that G H (ξ, η) > 0 (even if ξ = η) since ξ, η ∈ H are interior points.

∀ξ ∈ ∂Ω k , η ∈ Ω k : ∇ ξ ∇ η G k (ξ, η) = 0, ∇ 2 η G k (ξ, η) = 0 and of ∀ξ ∈ Ω k , η ∈ ∂Ω k : ∇ ξ ∇ η G k (ξ, η) = 0, ∇ 2 ξ G k (ξ, η) =
Lemma 7 (Estimates in the interior). Suppose that n = 2 and that Ω ⊂ R 2 is a bounded C 4,γ -smooth domain. Then for each x 0 ∈ Ω there exists a radius r = r x0 > 0 and a constant C = C x0 > 0 such that for all x, y ∈ Ω x0,r := Ω∩B r (x 0 ) one has [START_REF] Ju | Isolation of singularities of the Green's function (Russian)[END_REF] G

Ω (x, y) ≥ Cd(x)d(y) min 1, d(x)d(y) |x -y| 2 .
Proof. Since x 0 ∈ Ω, we have that G Ω (x 0 , x 0 ) > 0 (see [17, p. 115]): since G Ω is continuous, there exists r, c > 0 such that B r (x 0 ) ⊂⊂ Ω and G Ω (x, y) > c for all x, y ∈ B r (x 0 ). This yields [START_REF] Ju | Isolation of singularities of the Green's function (Russian)[END_REF] since Ω is bounded. we find:

Corollary 8. Suppose that n = 2 and that Ω ⊂ R 2 is a bounded C 4,γ -smooth domain. Then there exists a radius r > 0 and a constant C > 0 such that for all x 0 ∈ Ω and for all x, y ∈ Ω x0,r := Ω ∩ B r (x 0 ) one has

(17) G Ω (x, y) ≥ Cd(x)d(y) min 1, d(x)d(y) |x -y| 2 .
This step provides in particular a different and simpler proof for the local positivity statement from [9, Theorem 6.15] which was proved first by Dall'Acqua, Meister, and Sweers [START_REF] Dall'acqua | Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains[END_REF]. Proof of Theorem 1: If |x -y| ≥ r we take from [START_REF] Dall'acqua | Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems[END_REF], see also [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF], that G

Ω (x, y) ≥ -c 8 |x -y| -2 d(x) 2 d(y) 2 so that G Ω (x, y) + 2c 8 |x -y| -2 d(x) 2 d(y) 2 ≥ c 8 |x -y| -2 d(x) 2 d(y) 2 ≥ c 8 d(x)d(y) min 1, d(x)d(y) |x -y| 2 .
Since |x -y| -2 ≤ r -2 we end up with

G Ω (x, y) + c 9 d(x) 2 d(y) 2 ≥ c 8 d(x)d(y) min 1, d(x)d(y) |x -y| 2 .
and positive constants c 8 , c 9 > 0 in this case. The proof of Theorem 1 is now complete also for n = 2 using Corollary 8.

Proof of Theorem 2

This theorem was proved in [START_REF] Grunau | Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions[END_REF] when n ≥ 3 and in [START_REF] Gazzola | Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains[END_REF]Theorem 6.15] when n = 2. The latter proof is quite involved and based on the extensive use of conformal maps and explicit Green functions in certain limaçons. We provide here an alternate proof which uses the same techniques for n = 2 as for n ≥ 3. 2 ε 2 and we are in the situation just considered. So we are left with |x -y| < δn 2 ε and d(x) ≥ 1 2 ε. Then we may apply Theorem 1 of [START_REF] Grunau | Regions of positivity for polyharmonic Green functions in arbitrary domains[END_REF], see also Proposition 3, to find that for |x -y| < 1 2 δ n ε ≤ δ n max(d(x), d(y)) it follows that G Ω (x, y) > 0. For dimension n = 2 the result follows directly from Corollary 8.

( 1 ) 2 |x -y| 4 for

 124 -c d(x) 2 d(y) 2 ≤ G Ω (x, y) ≤ c * |x -y| 4-n min 1, d(x) 2 d(y) all x, y ∈ Ω ⊂ R n and where c, c * are some positive constants only depending on the domain. The distance of x to the boundary ∂Ω is defined by d(x) := inf {|x -x * | ; x * ∈ ∂Ω} , Date: March 3, 2011. 2000 Mathematics Subject Classification. Primary 35B51; Secondary 35J40, 35A08.

1

 1

  0 we obtain upon integration that|∇ 2 ξ,η G k (ξ, η)| ≤ C(1 + (log |ξ|) + + (log |η|) + + | log |ξ -η||) uniformly in k and further that |∇ ξ,η G k (ξ, η)| ≤ C(1 + (log |ξ|) + + (log |η|) + )(1 + |ξ| + |η|) uniformly in k; |G k (ξ, η)| ≤ C(1 + (log |ξ|) + + (log |η|) + )(1 + |ξ| 2 + |η| 2 ) uniformly in k.

5 .

 5 Proof of the main estimate for n = 2 Combining Lemmas 6 and 7 and applying a compactness argument to Ω = x0∈Ω Ω x0,rx 0 /2

  Case I: d(x)d(y) ≤ |x -y|2 . For this situation we have(18) H Ω (x, y) y| -n d(x) 2 d(y) 2 if n = 4, log 1 + d(x) 2 d(y) 2 |x -y| 4 if n = 4.Then there is c > 0 such that we find c -12 H Ω (x, y) ≥ c 1 d(x) 2 d(y) 2 for |x -y| < c. Case II: d(x)d(y) > |x -y| 2 . Now we have (19) H Ω (x, y) = d(x) 2 d(y) 2 |x -y| 4 if n = 4, d(x) 2-n/2 d(y)2-n/2 if n = 2, 3. Since d(x)d(y) is bounded on Ω one finds for n ≥ 4 the existence of c > 0 such that c -1 2 H Ω (x, y) ≥ c 1 d(x) 2 d(y) 2 for |x -y| < c. For dimension n = 3 the argument is more subtle. We fix δ n as in Proposition 3. Taking d(x)d(y) < ε 2 1+δn 2 for ε > 0 but sufficiently small it follows that c -1 2 H Ω (x, y) ≥ c 1 d(x) 2 d(y) 2 . It remains to consider d(x)d(y) ≥ 1+δn 2 ε 2 . Assume first that |x -y| < δn 2 ε and d(x) < 1 2 ε. Then d(x)d(y) < 1+δn


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