Mahamadi Warma
email: warma@uprrp.edu

DIRICHLET AND NEUMANN BOUNDARY CONDITIONS FOR THE p-LAPLACE OPERATOR: WHAT IS IN BETWEEN?

Keywords: 1991 Mathematics Subject Classification. Primary 35J65, 35K55, 47H20 Local functionals, integral representation, subgradients, Robin boundary conditions, nonlinear semigroups, domination of semigroups. 1

Let p ∈ (1, ∞) and let Ω ⊆ R N be a bounded domain with Lipschitz continuous boundary. We characterize all order preserving semigroups on L 2 (Ω) which are generated by convex, lower semicontinuous, local functionals and which are sandwiched between the semigroups generated by the p-Laplace operator with Dirichlet and Neumann boundary conditions. We show that every such semigroup is generated by the p-Laplace operator with Robin type boundary conditions.

INTRODUCTION

Let Ω ⊆ R N be a bounded domain with Lipschitz continuous boundary ∂ Ω. It is well known that for every p ∈ (1, ∞) the diffusion equation governed by the p-Laplace operator (1.1) The two problems can be rewritten as abstract gradient systems of the form u + ∂ ϕ(u) 0 on R + , u(0) = u 0 , with ϕ : L 2 (Ω) → [0, +∞] being a convex and lower semicontinuous functional and ∂ ϕ being its subgradient; the well-posedness of such gradient systems follows from a classical result by Minty. In the case of Dirichlet boundary conditions, this functional is

   u t -∆ p u = 0 in (0, ∞) × Ω, u(0, •) = u 0 in Ω,
(1.2) ϕ D (u) :=    1 p Ω |∇u| p dx, if u ∈ W 1,p 0 (Ω) ∩ L 2 (Ω), +∞ otherwise
and in the case of Neumann boundary conditions it is

(1.3) ϕ N (u) :=    1 p Ω |∇u| p dx, if u ∈ W 1,p (Ω) ∩ L 2 (Ω), +∞ otherwise.
The associated gradient systems give rise to strongly continuous semigroups of nonlinear (linear if p = 2) contractions on L 2 (Ω), denoted by S D and S N , respectively. It is well known that both semigroups are order preserving, and that S D is dominated by the semigroup S N in the sense that |S D (t)u| ≤ S N (t)|u| for every t ≥ 0 and every u ∈ L 2 (Ω).

In other words, the diffusion governed by the p-Laplace operator with Neumann boundary conditions (perfectly isolating boundary; the total energy Ω u is conserved) dominates the diffusion governed by the p-Laplace operator with Dirichlet boundary conditions (perfectly conducting boundary; the energy dissipates through the boundary).

We also consider the functional ϕ on L 2 (Ω) defined by

(1.4) ϕ(u) :=    1 p Ω |∇u| p dx + ∂ Ω B(x, u) dµ, if u ∈ D(ϕ), +∞ otherwise,
with effective domain

D(ϕ) = {u ∈ W 1,p (Ω) ∩ L 2 (Ω) : ∂ Ω B(x, u) dµ < ∞},
where µ is a regular Borel measure on ∂ Ω, and B : ∂ Ω × R → [0, +∞] is a Borel function which is measurable in the first variable, and lower semicontinuous and bi-monotone (that is, nonincreasing on ] -∞, 0] and nondecreasing on [0, +∞[) in the second variable. This functional is equal to ϕ D or to ϕ N , if µ = σ is the surface measure on ∂ Ω and if

B(x, s) =    +∞ if s = 0, 0 if s = 0,
in the case of Dirichlet boundary conditions, and B(x, s) = 0 in the case of Neumann boundary conditions. In any case, if ϕ is convex and lower semicontinuous, then ∂ ϕ is a realization of the p-Laplace operator with generalized Robin type boundary conditions formally given by (1.5) |∇u| p-2 ∂ u/∂ νdσ + β (x, u)dµ 0 on ∂ Ω.

If µ = σ or, more generally, if µ is absolutely continuous with respect to σ (hence, dµ = α(x) dσ ), then (1.5) reduces to the classical Robin boundary conditions. For more details on this formulation of the boundary conditions, we refer to [START_REF] Arendt | The Laplacian with Robin boundary conditions on arbitrary domains[END_REF][START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF][START_REF] Biegert | The heat equation with nonlinear generalized Robin boundary conditions[END_REF][START_REF] Biegert | Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains[END_REF][START_REF] Biegert | Existence of solutions to quasi-linear elliptic problems with generalized Robin boundary conditions[END_REF][START_REF] Daners | Robin boundary value problems on arbitrary domains[END_REF][START_REF] Daners | A priori estimates for a class of quasi-linear elliptic equations[END_REF] and the references therein, and to Section 3.1 below. The subgradient ∂ ϕ generates a nonlinear semigroup S of contractions on L 2 (Ω). Under appropriate further conditions on µ and B (see Theorem 2.1 for the precise conditions), the semigroup S is sandwiched between the semigroups S D and S N in the sense that (1.6) |S D (t)u| ≤ S(t)|u| and |S(t)u| ≤ S N (t)|u| for every t ≥ 0 and u ∈ L 2 (Ω).

In this sense, the diffusion governed by the p-Laplace operator with Robin type boundary conditions is intermediate between the diffusions governed by the p-Laplace operator with Dirichlet and Neumann boundary conditions.

The aim of this paper is to prove a converse of (1.6). More precisely, we show that if S is a semigroup on L 2 (Ω) generated by the subgradient of a convex and lower semicontinuous functional ϕ on L 2 (Ω), if S is sandwiched between S D and S N in the sense of (1.6), and if ϕ satisfies a natural locality condition (so that its subgradient is a local operator), then ϕ is necessarily of the form (1.4) for some µ and some B (Theorem 2.1). This is a possible answer to the question Dirichlet and Neumann boundary condition: What is in between? which was asked in the title of the article by Arendt & Warma [START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF]. The article [START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF] gave an answer to this question in the linear case, that is, in the case of the diffusion governed by the Laplace operator and when all semigroups are C 0 -semigroups of linear, selfadjoint operators. Our article (and the title of our article) is clearly motivated by the question considered in [START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF], and by the question whether a similar characterization of sandwiched semigroups holds in the context of a nonlinear diffusion equation.

We outline the plan of the paper. In Section 2 we give some preliminaries and state the main result of this article (Theorem 2.1). Concerning the proof of the main result, we follow in some sense the idea of proof of the corresponding result in the linear case [START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF]Theorem 4.1]. Our proof thus heavily depends on a characterization of domination and order preservation of semigroups in terms of properties of the generating functionals and on a Riesz type representation theorem for convex, lower semicontinuous functionals. These two results are also stated in Section 2. Characterizations of domination and order preservation of nonlinear semigroups generated by subdifferentials go back to Brézis & Pazy [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF], but the formulation which is appropriate for our purposes (Theorem 2.2) is taken from Barthélemy [START_REF] Barthélemy | Invariance d'un convex fermé par un semi-groupe associé à une forme non-linéaire[END_REF]. There exist also several Riesz type representation theorems for nonlinear functionals (see, for example, [START_REF] Maso | Asymptotic behaviour of minimum problems with bilateral obstacles[END_REF][START_REF] Maso | Integral representation for a class of C 1 -convex functionals[END_REF][START_REF] Maso | Γ-limits of obstacles[END_REF][START_REF] Marcus | Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces[END_REF]), but no appropriate result is stated for lower semicontinuous functionals on the Sobolev space W 1,p (Ω). Therefore, we state and prove such a result (Theorem 2.3) which seems to be new to the best of our knowledge and may have its own, independent interest. Sections 4 and 5 are devoted to the proofs of Theorems 2.1 and 2.3, while in Section 3 we include a discussion to clarify the conditions from the main theorem.

MAIN RESULT

Let H be a real Hilbert space with inner product (•, •) H , and let ϕ : H → (-∞, +∞] be a convex and lower semicontinuous (l.s.c.) functional with effective domain

D(ϕ) := {u ∈ H : ϕ(u) < ∞}.
By a classical result of Minty [START_REF] Minty | Monotone (nonlinear) operators in Hilbert space[END_REF] (see also Minty [25] or the monographs by Brézis [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF], Evans [START_REF] Evans | Partial Differential Equations[END_REF], or Showalter [START_REF] Showalter | Monotone Operators in Banach Space and Nonlinear Partial Differential Equations[END_REF]), every convex, l.s.c. functional ϕ on H generates a strongly continuous semigroup S = (S(t)) t≥0 of (in general nonlinear) contractions on D(ϕ). This means that there exists a unique family S = (S(t)) of contractions on D(ϕ) such that for every u 0 ∈ D(ϕ) the trajectory u := S(•)u 0 is the unique strong solution of the abstract gradient system

   u ∈ C(R + ; H) ∩W 1,∞ loc ((0, ∞); H), u + ∂ ϕ(u) 0 almost everywhere on R + , u(0) = u 0 .
Here, the subgradient ∂ ϕ at a point u ∈ D(ϕ) is defined by

∂ ϕ(u) := { f ∈ H : ϕ(u + w) -ϕ(u) ≥ ( f , w) H for every w ∈ H}.
In fact, throughout the following, H = L 2 (Ω) for some bounded domain Ω ⊆ R N with Lipschitz continuous boundary, and all functionals on L 2 (Ω) have dense effective domain, unless otherwise stated. The space L 2 (Ω) is a real Hilbert lattice for the natural ordering. It thus makes sense to consider the following properties of a semigroup or a pair of semigroups. We say that a semigroup S = (S(t)) t≥0 on L 2 (Ω) is order preserving, if (2.1) S(t)u ≤ S(t)v for all t ≥ 0 whenever u, v ∈ L 2 (Ω) and u ≤ v.

Moreover, if S 1 = (S 1 (t)) t≥0 and S 2 = (S 2 (t)) t≥0 are two semigroups on L 2 (Ω), then we say that S 1 is dominated by S 2 and we write S 1 S 2 , if

|S 1 (t)u| ≤ S 2 (t)|u| for all u ∈ L 2 (Ω) and t ≥ 0.
We say that a functional ϕ :

L 2 (Ω) → (-∞, +∞] is local 1 if for every u, v ∈ L 2 (Ω) (2.2) |u| ∧ |v| = 0 ⇒ ϕ(u + v) = ϕ(u) + ϕ(v).
Here, u ∧ v denotes the (pointwise) infimum of the functions u and v.

(ϕ D ) = W 1,p 0 (Ω) ∩ L 2 (Ω) and D(ϕ N ) = W 1,p (Ω) ∩ L 2 (Ω) are both dense in L 2 (Ω).
Hence, the semigroups S D and S N generated by the two functionals ϕ D and ϕ N are both defined on the whole space L 2 (Ω). The subgradients ∂ ϕ D and ∂ ϕ N are realizations of the p-Laplace operator with Dirichlet and Neumann boundary conditions, respectively. It is well known that the semigroups S D and S N are both order preserving, and that S D is dominated by S N (see [START_REF] Ouhabaz | Invariance of closed convex sets and domination criteria for semigroups[END_REF] for p = 2 and [START_REF] Barthélemy | Invariance d'un convex fermé par un semi-groupe associé à une forme non-linéaire[END_REF] for general p).

(2.3)

A set A ⊆ R N is called p-polar if Cap p (A) = 0. A statement P(x) is said to hold p-quasi everywhere on B ⊆ R N , if there exists a p-polar set A ⊆ R N such that the statement P(x) holds for every x ∈ B \ A. A function u : B → R (B ⊆ R N ) is said to be p-quasi continuous if for every ε > 0 there exists an open set O ⊆ R N such that Cap p (O) < ε and u restricted to B \ O is continuous. It is well known that every u ∈ W 1,p (Ω) admits a p-quasi continuous representative ũ : Ω → R. This p-quasi continuous representative is unique up to a p-polar set, that is, every two p-quasi continuous representatives coincide p-quasi everywhere on Ω. Throughout the following, we identify each function u ∈ W 1,p (Ω) with a

p-quasi-continuous representative. A subset G of R N is said to be p-quasi open if for every ε > 0 there exists an open set O ⊆ R N such that Cap p (O) < ε and G ∪ O is open.
Despite the fact that the p-capacity is not a Borel measure (the p-capacity is not σadditive), we say that the measure µ is absolutely continuous with respect to the p-capacity if for every p-polar Borel set A ⊆ Ω, one has µ(A) = 0.

Finally, a function B : R →] -∞, +∞] is called bi-monotone if it is nonincreasing on ] -∞, 0] and nondecreasing on [0, +∞[.

The following is the main theorem of this article. 1 In the literature, one can also find the term additive. (i) The semigroup S is local, order preserving and S D S S N . (ii) There exist a finite, regular Borel measure µ on ∂ Ω which is absolutely continuous with respect to the p-capacity, and a Borel function B :

∂ Ω × R → [0, +∞] satisfying (H)          B(•, s) is measurable for every s ∈ R, B(x, 0) = 0 for µ -a.e. x ∈ ∂ Ω, B(x, •) is lower semicontinuous for µ -a.e. x ∈ ∂ Ω B(x, •) is bi-monotone for µ -a.e. x ∈ ∂ Ω such that ϕ(u) = 1 p Ω |∇u| p dx + ∂ Ω B(x, u) dµ for all u ∈ D(ϕ).
The proof of Theorem 2.1 is carried out in Section 5, while Section 3 is reserved for a discussion of Theorem 2.1. The proof of the implication (ii)⇒(i) is based on the following theorem and relatively straightforward. The following theorem due to Barthélemy [3] is an application, using also clever convexity arguments, of the characterization that a semigroup generated by a convex, l.s.c. functional ϕ : L 2 (Ω) → [0, +∞] leaves a closed, convex set invariant. This characterization goes back to Brézis & Pazy [START_REF] Brézis | Semigroups of nonlinear contractions on convex sets[END_REF]; see also Brézis [9, Section IV.4], Cipriano & Grillo [START_REF] Cipriani | Nonlinear Markov semigroups, nonlinear Dirichlet forms and application to minimal surfaces[END_REF] and the references therein. We state the theorem for densely defined functionals only.

Theorem 2.2 (Barthélemy). Let ϕ, ϕ 1 , ϕ 2 : L 2 (Ω) → (-∞, +∞] be three convex and l.s.c. functionals with dense effective domains. Let S, S 1 , and S 2 be the semigroups generated by ϕ, ϕ 1 , and ϕ 2 , respectively. Then:

(a) ([3, Théorème 2.1]) If the functional ϕ is nonnegative, then the semigroup S is order preserving if and only if for all u, v ∈ L 2 (Ω) one has

(2.4) ϕ (u ∧ v) + ϕ (u ∨ v) ≤ ϕ(u) + ϕ(v). (b) ([3, Théorème 3.3]) If
, v ∈ L 2 (Ω), v ≥ 0, (2.5) ϕ 1 ((|u| ∧ v) • sgn(u)) + ϕ 2 (|u| ∨ v) ≤ ϕ 1 (u) + ϕ 2 (v).
The difficult part in Theorem 2.1 is the implication (i)⇒(ii). Its proof uses the above characterization of order preservation and domination, too, but it uses in addition the following Riesz type representation theorem which may have its own interest, independently of the application given in this article. Similar representation theorems for various classes of functionals on various functions spaces are included in [START_REF] Chacon | Additive functionals[END_REF][START_REF] Chill | A Riesz type representation for lower semicontinuous, monotone, local functionals on C c (X) +[END_REF][START_REF] Maso | Asymptotic behaviour of minimum problems with bilateral obstacles[END_REF][START_REF] Maso | Integral representation for a class of C 1 -convex functionals[END_REF][START_REF] Maso | Γ-limits of obstacles[END_REF][START_REF] Friedman | A representation theorem for additive functionals[END_REF][START_REF] Friedman | On additive functionals[END_REF][START_REF] Marcus | Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces[END_REF][START_REF] Rao | Local functionals. Measure theory[END_REF][START_REF] Tihonov | On some non-linear functionals[END_REF] and the references therein.

Let p ∈ (1, ∞). We denote by W 1,p (Ω) + the positive cone in W 1,p (Ω). Given a functional ψ : W 1,p (Ω) + → [0, +∞], we call D(ψ) = {u ∈ W 1,p (Ω) + : ψ(u) < +∞} its effective domain. The effective support of the functional ψ is the set supp[ψ] := Ω \ {x ∈ Ω : there exists a neighborhood U of x such that for every

u ∈ D(ψ) with supp[u] ⊆ U one has ψ(u) = 0}.
We say that the functional

ψ is monotone if for every u, v ∈ W 1,p (Ω) + (2.6) u ≤ v ⇒ ψ(u) ≤ ψ(v).
Similarly as for functionals defined on L 2 (Ω) we say that

ψ is local if for every u, v ∈ W 1,p (Ω) + (2.7) u ∧ v = 0 ⇒ ψ(u + v) = ψ(u) + ψ(v).
Theorem 2.3. Let Ω ⊆ R N be a bounded domain with Lipschitz continuous boundary. For every functional ψ : W 1,p (Ω) + → [0, +∞] the following assertions are equivalent.

(i) The functional ψ is lower semicontinuous, monotone, local and for every u, v ∈ D(ψ), one has u ∨ v, u ∧ v ∈ D(ψ) and

(2.8) ψ(u ∨ u) + ψ(u ∧ v) ≤ ψ(u) + ψ(v).
(ii) There exist a finite, regular Borel measure µ with supp[µ] ⊆ supp[ψ] which is absolutely continuous with respect to the p-capacity, and a Borel function

B : Ω × R + → [0, +∞] satisfying (H + )          B(•, s) is measurable for every s ∈ R, B(x, 0) = 0 for µ -a.e. x ∈ Ω, B(x, •) is lower semicontinuous for µ -a.e. x ∈ Ω, B(x, •) is monotone for µ -a.e. x ∈ Ω such that ψ(u) = Ω B(x, u) dµ for all u ∈ D(ψ).
Section 4 is devoted to the proof of this theorem.

ψ i (u) = Ω B i (x, u) dµ for every u ∈ D(ψ i ) (i = 1, 2)
. It suffices to take for example µ = µ 1 + µ 2 , where µ 1 and µ 2 are two representing measures for ψ 1 and ψ 2 , respectively, the existence of which is guaranteed by Theorem 2.3. (c) Theorem 2.3 remains true if the space W 1,p (Ω) is replaced by the a priori smaller space W 1,p (Ω) ∩ L 2 (Ω). This is trivially true for p ≥ 2, since then the two spaces actually coincide. If p < 2, essentially the same proof works. At a first glance, it seems necessary to replace everywhere the p-capacity by the following (p, 2)-capacity which is for subsets

A ⊆ R N defined by Cap (p,2) (A) = inf u p W 1,p (R N ) + u 2 L 2 (R N ) : u ∈ W 1,p (R N ) ∩ L 2 (R N
) and there exists

O ⊆ R N open, such that A ⊆ O and u ≥ 1 a.e. on O .
However, it is actually not necessary to do this since for p ≤ 2 the p-capacity and the (p, 2)-capacity are equivalent in the sense that for every subset

A ⊆ R N Cap p (A) ≤ Cap (p,2) (A) ≤ 2 Cap p (A).
Here, the first inequality is obvious from the respective definitions of the two capacities, while for the second inequality one has to notice that the definition of the (p, 2)-capacity does not change if one takes the infimum over functions satisfying 0 ≤ u ≤ 1 everywhere (and u = 1 a.e. on O) and that u 2

L 2 (R N ) ≤ u p L p (R N ) whenever 0 ≤ u ≤
(x, •) is convex for µ-a.e. x ∈ ∂ Ω. Denote by β (x, •) = ∂ B(x, •) the subgradient of the functional B(x, •), that is, for s ∈ D(B(x, •)), β (x, s) = {τ ∈ R : B(x, s + ξ ) -B(x, s) ≥ τξ for every ξ ∈ R}. Let f ∈ L 2 (Ω). We say that a function u ∈ W 1,p (Ω) ∩ L 2 (Ω) is a weak solution of the elliptic problem (3.1)    -∆ p u = f in Ω, |∇u| p-2 ∂ u ∂ ν dσ + β (x, u) dµ 0 on ∂ Ω, if -∆ p u = f in the sense of distributions, if ∂ Ω B(x, u) dµ < +∞ and if for every w ∈ W 1,p (Ω) ∩ L 2 (Ω) (3.2) ∂ Ω B(x, u + w) -B(x, u) dµ ≥ Ω f w dx - Ω |∇u| p-2 ∇u∇w dx.
The relation between this inequality and the boundary condition in (3.1) becomes clear if one replaces f by -∆ p u, if one recalls Green's formula

Ω ∆ p u w dx + Ω |∇u| p-2 ∇u ∇w dx = ∂ Ω |∇u| p-2 ∂ u ∂ ν w dσ (w ∈ W 1,p (Ω))
(which holds for sufficiently smooth functions u), and if one uses the definition of the subgradient.

Let ϕ be as in Theorem 2.1 (ii), that is,

ϕ(u) =    1 p Ω |∇u| p dx + ∂ Ω B(x, u) dµ if u ∈ D(ϕ) +∞ otherwise,
where the effective domain is given by

D(ϕ) = {u ∈ W 1,p (Ω) ∩ L 2 (Ω); ∂ Ω B(x, u) dµ < ∞}. Proposition 3.1. Let f ∈ L 2 (Ω). Then u ∈ W 1,p (Ω) ∩ L 2 (Ω) is a weak solution of (3.1) if and only if u ∈ D(ϕ) and f ∈ ∂ ϕ(u).
Proof. Assume that u ∈ D(ϕ) and f ∈ ∂ ϕ(u). Then, by the definition of D(ϕ), u ∈ W 1,p (Ω) ∩ L 2 (Ω), ∂ Ω B(x, u) dµ < +∞, and, as a consequence of the definition of the subgradient ∂ ϕ, for every w ∈ W 1,p (Ω) ∩ L 2 (Ω)

1 p Ω (|∇(u + w)| p -|∇u| p ) dx + ∂ Ω (B(x, u + w) -B(x, u)) dµ ≥ Ω f w dx.
In particular, when we replace w by tw (t > 0), divide the inequality by t and let t tend to 0, we obtain that for every w ∈ W 1,p (Ω) ∩ L 2 (Ω)

(3.3) Ω |∇u| p-2 ∇u∇w dx + lim inf t→0+ ∂ Ω B(x, u + tw) -B(x, u) t dµ ≥ Ω f w dx.
In particular, for every test function w ∈ D(Ω),

Ω |∇u| p-2 ∇u∇w dx ≥ Ω f w dx.
Since this inequality is true for w and -w (w ∈ D(Ω)), one actually has equality, and therefore -∆ p u = f in the sense of distributions. Finally, by convexity of B(x, •), the inequality (3.3) holds for every w ∈ W 1,p (Ω) ∩ L 2 (Ω) if and only if the inequality (3.2) holds for every w ∈ W 1,p (Ω) ∩ L 2 (Ω). Hence, u is a weak solution of (3.1). Conversely, let u ∈ W 1,p (Ω) ∩ L 2 (Ω) be a weak solution of (3.1). Then, by definition of weak solution and by definition of D(ϕ), u ∈ D(ϕ). Moreover, the inequality (3.2) holds for every

w ∈ W 1,p (Ω) ∩ L 2 (Ω), that is ϕ(u + w) -ϕ(u) ≥ Ω f w dx for every w ∈ W 1,p (Ω) ∩ L 2 (Ω).
Clearly this inequality holds trivially for w ∈ L 2 (Ω)\W 1,p (Ω), too. Hence, f ∈ ∂ ϕ(u).

3.2.

Regularity of Ω. Theorems 2.1 and 2.3 remain true if Ω is a bounded domain with the W 1,p -extension property (for more details on the extension property we refer to [START_REF] Maz'ya | Sobolev Spaces[END_REF][START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF]). This means that there exists a bounded, linear extension operator W 1,p (Ω) → W 1,p (R N ). We note that the W 1,p -extension property depends on p. If Ω has Lipschitz continuous boundary ∂ Ω, as we assumed in Theorems 2.1 and 2.3, then Ω has the W 1,p -extension property for every p ∈ (1, ∞), [29, Chap VI Theorem 5, p.181]. The Lipschitz continuity of the boundary (or: the W 1,p -extension property) is important in several places where we deal with the p-capacity. Note that the p-capacity is defined by means of W 1,p (R N ) functions.

If Ω does not have a Lipschitz continuous boundary, but is just an arbitrary bounded, open set, one may replace the p-capacity by the relative p-capacity, and replace the Sobolev space W 1,p (Ω) by the space

W 1,p (Ω) = W 1,p (Ω) ∩C( Ω) W 1,p (Ω)
. The relative p-capacity Cap p,Ω is for subsets A ⊆ Ω defined by see [START_REF] Arendt | The Laplacian with Robin boundary conditions on arbitrary domains[END_REF] for the case p = 2 and [START_REF] Biegert | Elliptic problems on varying domains[END_REF][START_REF] Biegert | Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains[END_REF] for general p ∈ [1, ∞[. Up to these changes, and up to replacing the functional ϕ N (and the associated semigroup S N ) by the functional

φN (u) :=    1 p Ω |∇u| p dx, if u ∈ W 1,p (Ω) ∩ L 2 (Ω),
+∞ otherwise (and the associated semigroup SN ), the main results in this article (Theorems 2.1 and 2.3) hold with essentially the same proofs. shows that these regularity conditions may be dropped.

In the general situation of Theorem 2.1, we are actually not able to show that D(ϕ) ∩ C( Ω) is dense in D(ϕ) (in the W 1,p (Ω) topology, for example). We are even not sure whether D(ϕ) contains nontrivial continuous functions at all. This problem is also the reason why we can not use a Riesz type representation theorem for functionals defined on C( Ω) + such as for example the representation theorems in [START_REF] Chill | A Riesz type representation for lower semicontinuous, monotone, local functionals on C c (X) +[END_REF] or [START_REF] Tihonov | On some non-linear functionals[END_REF]. Instead, we are forced to use a Riesz type representation theorem on W 1,p (Ω) + (that is, Theorem 2.3).

If in the situation of Theorem 2.1 one assumes in addition that the functional ϕ is continuous on W 1,p (Ω), then in Theorem 2.3 one obtains that B(x, •) is continuous. In this situation, it is possible to show that the functional ϕ is also regular in the sense that for every u ∈ D(ϕ) there exists a sequence

(u n ) ⊆ D(ϕ) ∩ C( Ω) converging to u in W 1,p (Ω) and satisfying lim n→∞ ϕ(u n ) = ϕ(u).
We finally note that if p = 2 in Theorem 2.1, that is the case of the Laplace operator, even if S D and S N are linear semigroups, Theorem 2.1 shows that there are nonlinear sandwiched semigroups generated by the Laplace operator with nonlinear boundary conditions of the form ∂ u/∂ νdσ + β (x, u)dµ 0.

3.4. Nonnegativity of the functional ϕ. While the generation theorem by Minty applies for general convex, l.s.c. functionals ϕ : L 2 (Ω) →] -∞, +∞], the functional in Theorem 2.1 is assumed to be convex, l.s.c. and nonnegative. However, assuming condition (i) in Theorem 2.1, ϕ is automatically nonnegative. In fact, the domination S S N implies that 0 is an equilibrium point of the semigroup S (that is, S(t)0 = 0 for every t ≥ 0). This is equivalent to 0 ∈ ∂ ϕ(0). Since ϕ is convex, this in turn is equivalent to the fact that ϕ attains its minimum in 0. Now, the further assumption that ϕ is local implies ϕ(0) = 0. Hence, ϕ is nessecarily nonnegative.

(3.4) ϕ(u) =    1 p Ω |∇u| p dx + ∂ Ω B(x, u) dµ, if u ∈ D(ϕ) +∞ otherwise,
where the effective domain

D(ϕ) := {u ∈ W 1,p (Ω) ∩ L 2 (Ω) : ∂ Ω B(x, u) dµ < ∞}.
Theorem 3.2. If the functional ϕ defined above is lower semicontinuous on L 2 (Ω), then for every p-polar set K ⊆ ∂ Ω and every u ∈ D(ϕ) one has K B(x, u) dµ = 0.

Proof. Let K ⊆ ∂ Ω be a p-polar set, and let u ∈ D(ϕ). We first assume that K is compact and that u is nonnegative and essentially bounded. Since K is p-polar, there exists a sequence By the bounded convergence theorem, we also have lim n→∞ uu n L 2 (Ω) = 0. Since ϕ is lower semicontinuous on L 2 (Ω), we obtain

(v n ) ⊆ W 1,p (R N ) such that 0 ≤ v n ≤ u L ∞ (Ω) everywhere on Ω, v n = u L ∞ (Ω) on K,
1 p Ω |∇u| p + ∂ Ω B(x, u) dµ ≤ lim inf n→∞ 1 p Ω |∇(u -u n )| p + ∂ Ω B(x, u -u n ) dµ .
The inequality uu n ≤ u (everywhere on Ω), the equality uu n = 0 on K, the assumption B(x, 0) = 0, and the bi-monotonicity of B imply that for every n

∂ Ω B(x, u -u n ) dµ ≤ ∂ Ω\K B(x, u) dµ.
The preceding two inequalities and the convergence u n W 1,p (Ω) → 0 together imply K B(x, u) dµ ≤ 0.

Since B ≥ 0, we thus obtain K B(x, u) dµ = 0. The equality K B(x, u) dµ = 0 for arbitrary nonnegative u ∈ D(ϕ) (but compact K) follows by an approximation with the sequence (u ∧ n), using also the lower semi-continuity of ϕ and the monotonicity of B. If K is not compact (but p-polar) and if u ∈ D(ϕ) is nonnegative, then we obtain the equality K B(x, u) dµ = 0 from the inner regularity of the measure B(x, u) dµ.

Similarly, using the bi-monotonicity of B, one shows that K B(x, u) dµ = 0 for every p-polar set K ⊆ ∂ Ω and every nonpositive u ∈ D(ϕ). Finally, if K is an arbitrary p-polar set and u ∈ D(ϕ) is arbitrary, too, then the previous steps imply

K B(x, u) dµ = K B(x, u + -u -) dµ = K B(x, u + ) dµ + K B(x, -u -) dµ = 0,
where we have also used the fact if u ∈ D(ϕ), then u + ∈ D(ϕ).

With a slight abuse of language, Theorem 3.2 says that if the functional ϕ given by (3.4) is lower semicontinuous, then the weighted measure B dµ is necessarily absolutely continuous with respect to the p-capacity (in the sense made precise in Theorem 3.2). We point out that the stronger property that the unweighted measure µ is absolutely continuous with respect to the p-capacity can not be expected in the general situation of Theorem 3.2 (take, for example, B = 0 and µ a measure which is not absolutely continuous with respect to the p-capacity). At the same time, we point out that our main Theorem 2.1 does state the existence of a representing measure µ which is absolutely continuous with respect to the p-capacity.

In the literature (see, for example, [START_REF] Biegert | The heat equation with nonlinear generalized Robin boundary conditions[END_REF][START_REF] Biegert | Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains[END_REF][START_REF] Biegert | Existence of solutions to quasi-linear elliptic problems with generalized Robin boundary conditions[END_REF][START_REF] Daners | A priori estimates for a class of quasi-linear elliptic equations[END_REF] for the nonlinear case and [START_REF] Arendt | The Laplacian with Robin boundary conditions on arbitrary domains[END_REF][START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF][START_REF] Daners | Robin boundary value problems on arbitrary domains[END_REF] for the linear case) parabolic and elliptic equations associated with the functional ϕ defined in (3.4) have been investigated. There, the authors have always assumed that B(x, •) is convex for µ-a.e. x ∈ ∂ Ω, and that the measure µ is absolutely continuous with respect to the p-capacity. Theorem 3.2 shows that this is a natural assumption (for obtaining wellposedness of the associated evolution problem, for example, but also for existence and regularity of weak solutions to associated elliptic problems).

The following result is a partial converse of Theorem 3.2. Theorem 3.3. Assume that Bdµ is absolutely continuous with respect to the p-capacity in the sense that for every p-polar set K ⊆ ∂ Ω and every u ∈ D(ϕ) one has K B(x, u) dµ = 0. If the functional ϕ given by (3.4) is convex, then ϕ is lower semicontinuous on L 2 (Ω).

Proof. We have to show that for every c ∈ R the set {ϕ ≤ c} is closed in L 2 (Ω). So fix c ∈ R. Let A be a closed (bounded) ball in L 2 (Ω) and let C := {ϕ ≤ c} ∩ A . Let (u n ) ⊆ C and u ∈ W 1,p (Ω) ∩ L 2 (Ω) be such that lim n→∞ u nu W 1,p (Ω)∩L 2 (Ω) = 0. The convergence in L 2 (Ω) implies that u ∈ A . The convergence in W 1,p (Ω) implies, after passing to a subsequence, that u n → u p-quasi-everywhere, that is, u n → u everywhere except possibly on a p-polar set K ⊆ ∂ Ω. Hence, by assumption on B and µ,

ϕ(u) = 1 p Ω |∇u| p + ∂ Ω B(x, u) dµ = 1 p Ω |∇u| p + ∂ Ω\K B(x, u) dµ ( K B(x, u) dµ = 0) ≤ 1 p Ω |∇u| p + ∂ Ω\K lim inf n→∞ B(x, u n ) dµ (B is l.s.c.) ≤ lim inf n→∞ 1 p Ω |∇u n | p + ∂ Ω\K B(x, u n ) dµ (by Fatou's lemma) = lim inf n→∞ 1 p Ω |∇u n | p + ∂ Ω B(x, u n ) dµ ( K B(x, u n ) dµ = 0) = lim inf n→∞ ϕ(u n ).
This shows u ∈ {ϕ ≤ c}. In particular, we have shown that C is closed in W

PROOF OF THEOREM 2.3

In this section, we prove Theorem 2.3. We start by proving the implication in Theorem 2.3 which is relatively straightforward.

Proof of Theorem 2.3, (ii) ⇒ (i). Assume that assertion (ii) holds. The monotonicity of the function B(x, •) (for µ-almost every x ∈ Ω, assumption (H + )) and the monotonicity of the integral imply that the functional ψ is monotone.

We show that ψ is lower semicontinuous. Let (u n ) ⊆ W 1,p (Ω) + be a sequence which converges to u ∈ W 1,p (Ω) + . By considering a subsequence, if necessary, we may assume that (u n ) converges to u p-quasi-everywhere, that is, there exists a p-polar set A ⊆ Ω such that (u n ) converges to u everywhere on Ω \ A (where possibly A is a larger p-polar set). Since, for µ-almost every x ∈ Ω, the function B(x, •) is lower semicontinuous, we obtain B(x, u(x)) ≤ lim inf n→∞ B(x, u n (x)) for every x ∈ Ω \ A. Using Fatou's lemma and the fact that, by assumption, the measure µ is absolutely continuous with respect to Cap p (this implies that K B(x, u) dµ = 0 for every p-polar set K ⊆ Ω and every u ∈ D(ψ)), we therefore obtain that

ψ(u) = Ω B(x, u) dµ = Ω\A B(x, u) dµ ≤ Ω\A lim inf n→∞ B(x, u n (x)) dµ ≤ ≤ lim inf n→∞ Ω\A B(x, u n (x)) dµ = lim inf n→∞ Ω B(x, u n (x)) dµ = lim inf n→∞ ψ(u n ). Hence, ψ is lower semicontinuous. Let u, v ∈ D(ψ) ⊆ W 1,p (Ω) + . Clearly, u ∨ v, u ∧ v ∈ W 1,p (Ω) + . From the equality ψ(u ∨ v) + ψ(u ∧ v) = Ω B(x, u ∨ v) dµ + Ω B(x, u ∧ v) dµ = {u≤v} B(x, v) dµ + {u≤v} B(x, u) dµ + {u>v} B(x, u) dµ + {u>v} B(x, v) dµ = Ω B(x, u) dµ + Ω B(x, v) dµ = ψ(u) + ψ(v)
we obtain that u ∨ v, u ∧ v ∈ D(ψ) and that (2.8) holds (even with equality).

Finally, let u, v ∈ W 1,p (Ω) + be such that u

∧ v = 0. Then u = v = 0 on supp[u] ∩ supp[v]. Since B(x, 0) = 0, we obtain that ψ(u + v) = Ω B(x, u + v) dµ = supp[u] B(x, u) dµ + supp[v] B(x, v) dµ - supp[u]∩supp[v] B(x, u + v) dµ = supp[u] B(x, u) dµ + supp[v] B(x, v) dµ = Ω B(x, u) dµ + Ω B(x, v) dµ = ψ(u) + ψ(v).
Hence, ψ is local.

To prove the converse implication (i)⇒(ii), we proceed stepwise, in the form of several lemmas.

Throughout the following, we denote by B the Borel σ -algebra of Ω. The set of all compact subsets of Ω is denoted by K . We assume also that the functional ψ satisfies the condition (i) in Theorem 2.3.

For δ > 0 and every subset K ⊆ Ω we define

K δ := {x ∈ Ω : d(x, K) ≤ δ }.
With this definition, for every compact subset K ∈ K we define

R(K) := {ρ ∈ W 1,∞ (R N ) : there exists δ > 0 such that ρ ≥ 1 on K δ }.
Note that every function ρ ∈ W 1,∞ (R N ) admits a Lipschitz continuous representative; the condition ρ ≥ 1 is to be understood as a pointwise inequality everywhere for this unique representative.

µ u (K) = inf ρ∈R(K) ψ(uρ) (K ∈ K ).
We remark that for every ρ ∈ R(K) one has ρ ∧ 1 ∈ R(K). Therefore, in the definition of µ u (K) it suffices to take the infimum over all functions ρ ∈ R(K) satisfying 0 ≤ ρ ≤ 1 everywhere and ρ = 1 on some K δ . In particular, by the monotonicity of ψ, µ u (K) is finite for every compact K ⊆ Ω.

Lemma 4.2 (Finite additivity). Let u ∈ D(ψ), and let K

1 , K 2 ⊆ Ω be two compact sets such that K 1 ∩ K 2 = / 0. Then µ u (K 1 ∪ K 2 ) = µ u (K 1 ) + µ u (K 2 ). Proof. Let K 1 , K 2 ⊆ Ω be two compact sets such that K 1 ∩ K 2 = / 0. Then d(K 1 , K 2 ) > 0. We can therefore find two functions ρ i ∈ R(K i ) (i = 1, 2) such that 0 ≤ ρ i ≤ 1 and ρ 1 ∧ ρ 2 = 0;
such a pair of functions ρ 1 , ρ 2 can easily be constructed by taking appropriate convolutions of characteristic functions and test functions.

Let ρ ∈ R(K 1 ∪ K 2 ). The monotonicity and locality of ψ implies

ψ(uρ) ≥ ψ(uρ(ρ 1 + ρ 2 )) = ψ(uρρ 1 ) + ψ(uρρ 2 ) ≥ µ u (K 1 ) + µ u (K 2 ). Since ρ ∈ R(K 1 ∪ K 2 ) was arbitrary, this implies µ u (K 1 ∪ K 2 ) ≥ µ u (K 1 ) + µ u (K 2 ). Lemma 4.7 (Monotonicity of µ u ). Let u, v ∈ D(ψ) be such that u ≤ v. Then µ u ≤ µ v .
Proof. The monotonicity of ψ and the definition of the measures µ u and µ v imply µ u (K) ≤ µ v (K) for every compact subset K ⊆ Ω. The claim then follows from the inner regularity of µ u and µ v .

Lemma 4.8. Let (u n ) ⊆ D(ψ) and u ∈ D(ψ) be such that u n ≤ u and lim n→∞ u n = u in W 1,p (Ω). Then

(4.3) lim n→∞ µ u n (G) = µ u (G) for every G ∈ B.
Proof. By Lemma 4.7, the domination

u n ≤ u implies µ u (G) -µ u n (G) ≥ 0 for every G ∈ B. Hence, for every G ∈ B, 0 ≤ lim sup n→∞ (µ u (G) -µ u n (G)) ≤ lim sup n→∞ µ u (G) -µ u n (G) + µ u (G c ) -µ u n (G c ) = lim sup n→∞ (µ u ( Ω) -µ u n ( Ω)) = lim sup n→∞ (ψ(u) -ψ(u n )) ≤ 0,
where in the last inequality we have used the lower semi-continuity of ψ. The preceding chain of inequalities implies the claim.

Lemma 4.9. For every u ∈ D(ψ) the measure µ u is absolutely continuous with respect to the p-capacity.

Proof. Let K ⊆ Ω be a p-polar set, that is, Cap p (K) = 0. We have to show that µ u (K) = 0. By inner regularity of µ u , we may assume K to be compact. Moreover, by replacing u by u ∧ n ∈ D(ψ) (with n ∈ N large enough), and by using Lemma 4.8, we see that we may in addition assume that u is essentially bounded. We assume both in the following.

Since Cap p (K) = 0, there exists a sequence (O n ) of open sets in R N and a sequence

(w n ) ⊆ W 1,p (R N ) ∩C c (R N ) such that 0 ≤ w n ≤ 1, w n = 1 on O n ⊇ K, and lim n→∞ w n W 1,p (R N ) = 0.
We claim that we can choose (O n ) and (w n ) such that

K ⊆ O n+1 ⊆ O n+1 ⊆ O n , K = n O n , and supp[w n+1 ] ⊆ O n .
First, observe that by replacing O n by the smaller set

O 1 ∩ • • • ∩ O n , we may assume that the sequence (O n ) is decreasing. Clearly K ⊆ n O n .
If there exists x ∈ n O n \ K, then we may replace O n by the smaller open set O n \ B(x, r), where r > 0 is sufficiently small so that one still has K ⊆ O n \ B(x, r) (recall that K is compact and that x has positive distance to K). In this way we can eliminate every x ∈ n O n \ K and finally have K = n O n . By using that K is compact and that the sets O n are open, it is straightforward to construct a subsequence of (O n ) (which we still denote by

O n ) such that K ⊆ O n+1 ⊆ O n+1 ⊆ O n . Now let (z k ) ⊆ W 1,∞ (R N ) be a sequence of functions satisfying 0 ≤ z k ≤ 1, z k = 1 on O k+1 and supp[z k ] ⊆ O k . For every k we can find m k ≥ k + 1 such that, for every m ≥ m k , z k w m W 1,p (R N ) ≤ 1 k . Note that, for every m ≥ m k , 0 ≤ z k w m ≤ 1, z k w m = 1 on O m , and supp[z k w m ] ⊆ O k . Now,
by replacing w n by z k n w m n for some appropriate sequences (k n ), (m n ) (so that k n+1 ≥ m n ), we obtain the desired claim.

By passing to a further subsequence, if necessary, we may finally assume that (w n ) and (∇w n ) converge pointwise almost everywhere to 0. Then it is straightforward to check that lim n→∞ uw n W 1,p (Ω) = 0. Moreover, for every n ∈ N,

ψ(u) ≥ ψ(u -uw n + uv n+1 ) (by monotonicity of ψ) = ψ(u -uw n ) + ψ(uv n+1 ) (by locality of ψ) (4.4) ≥ ψ(u -uw n ) + µ u (K)
(by definition of µ u ).

Since lim n→∞ (u-uw n ) = u and u-uw n ≤ u, the monotonicity and the lower semi-continuity of ψ imply lim

n→∞ ψ(u -uw n ) = ψ(u).
Hence, by passing to the limit in the inequality (4.4), we obtain ψ(u) ≥ ψ(u) + µ u (K), that is, µ u (K) = 0. Since K was an arbitrary p-polar set, this shows that µ u is absolutely continuous with respect to the p-capacity.

Lemma 4.10. For every u ∈ D(ψ) one has µ u ({u = 0}) = 0 (here {u = 0} denotes the null-set of a p-quasi-continuous representative of u; it is unique up to a p-polar set).

Proof. Let u ∈ D(ψ). By replacing u by u ∧ n ∈ D(ψ) (for n ∈ N sufficiently large) and by using Lemma 4.8, we see that we may assume that u is essentially bounded. This will be done in the following.

We show that we can find a sequence (u n ) ⊆ D(ψ) such that u n ≤ u, lim n→∞ u n = u in W 1,p (Ω) and u n = 0 in a neighborhood of {u = 0}. In fact, for every n ∈ N the set

n ) ⊆ W 1,p (R N ) such that 0 ≤ w n ≤ 1, w n = 1 p-quasi-everywhere on O n and lim n→∞ w n W 1,p (R N ) = 0. Let u n := (u ∧ n)(1 -w n ) and v n := (v ∧ n)(1 -w n ).
Then u n ≤ v n on the open set U n with u n = v n = 0 p-quasi-everywhere on O n . Moreover, u n ≤ u and v n ≤ v. Since U n is open, then for every compact set K ⊆ U ⊆ U n there exists δ > 0 such that K δ ⊆ U n . It follows from the monotonicity of ψ and the definition of the measures µ u n and µ v n that µ u n (K) ≤ µ v n (K) for every compact subset K ⊆ U. Passing to the limit and using Lemma 4.8, we obtain that µ u (K) ≤ µ v (K) for every compact set Proof. This lemma is an immediate consequence of Lemma 4.8, Lemma 4.10 and Lemma 4.11.

Recall that we identify each function w n ∈ D(ϕ) with a p-quasi-continuous representative; note that we may assume that this representative is nonnegative everywhere. For every x ∈ Ω we define the set W (x) := {w n (x) : n ∈ N}. Let I(x) be the closed convex hull of W (x), that is, the smallest, closed interval which contains W (x). Then, for every x ∈ Ω and every s ∈ R + we define Proof. For every s ∈ R + the set {x ∈ Ω : s ∈ I(x)} = {x ∈ Ω : s ≤ sup n w n (x)} is a Borel set. From this and from the definition of B one obtains that for every s ∈ R + the function

B(x, s) =    sup n B w n (x)1 {w n <s} (x) if s ∈ I(x), +∞ if s ∈ I(x).
B(•, s) is measurable.
It follows readily from the definition of B that B(x, 0) = 0 for every x ∈ Ω (since the sets {w n < 0} are empty and therefore 1 {w n <0} = 0 for every n). Moreover, since the sets {w n < s} are increasing with s ∈ R + , the function B(x, •) is monotone for every x ∈ Ω.

Finally, for every x ∈ Ω, every s ∈ I(x) \ {0} and every ε > 0 there exists, by definition of the supremum, n such that w n (x) < s and B(x, s)ε ≤ B w n (x) ≤ B(x, s). This implies B(x, s)ε ≤ B(x, s ) ≤ B(x, s) for every w n (x) < s ≤ s. As a consequence, B(x, •) is lower semicontinuous for every x ∈ Ω. Thus, B satisfies hypothesis (H + ) of Theorem 2.3.

Next, we show the second part of the statement. Let u ∈ D(ψ). By Lemma 4.14 (c), there exists a set A u of µ-measure zero such that for every n and every x ∈ {w n ≤ u} \ A u one has B w n (x) ≤ B u (x). As a consequence, B(•, u(•)) ≤ B u (•) µ-almost everywhere on Ω. In order to show the converse inequality we first note that, by Lemma 4.14 (b), B u (•) = 0 = B(x, 0) µ-almost everywhere on {u = 0}. Hence, it remains to show that the inequality B(•, u(•)) ≥ B u (•) holds almost everywhere on {u > 0}.

Let (λ m ) ⊆ [0, 1[ be a strictly increasing sequence such that lim m→∞ λ m = 1. Recall that the sequence (w n ) is dense in the effective domain D(ψ) (with respect to the W 1,p (Ω) topology) and that every sequence converging in W 1,p (Ω) admits a subsequence which converges p-quasi-everywhere. Hence, for every m there exists a subsequence (w α m (n) ) n such that lim n→∞ w α m (n) = λ m u p-quasi-everywhere. In particular,

u(x) > lim n→∞ w α m+1 (n) = λ m+1 u > λ m u p -quasi-everywhere on {u > 0}.
Since µ is absolutely continuous with respect to the p-capacity, the above convergence holds µ-almost everywhere. As a consequence, by Lemma 4.14 (c), for every m B(x, u(x)) = sup

ψ(u) = Ω B u (x) dµ(x) = Ω B(x, u(x)) dµ(x)
for every u ∈ D(ψ). Theorem 2.3 is completely proved.

PROOF OF THEOREM 2.1

In this section we give the proof of Theorem 2.1. We call a functional ψ :

W 1,p (Ω) → ] -∞, ∞] bi-monotone if for every u, v ∈ D(ψ) 0 ≤ u ≤ v ⇒ ψ(u) ≤ ψ(v) and u ≤ v ≤ 0 ⇒ ψ(u) ≥ ψ(v). (5.1)
Proof of Theorem 2.1, (ii) ⇒ (i). Let ϕ, µ and B be as in (ii).

Since B(x, 0) = 0 for µ-a.e. x ∈ ∂ Ω, then it is clear that the functional ϕ is local and hence, S is a local semigroup.

We show that S is order preserving. Since the functional ϕ is nonnegative, by Theorem 2.2 (b), it suffices to show that ϕ satisfies (2.4) for every u, v ∈ L 2 (Ω). Since the inequality (2.4) trivially holds if u or v does not belong to D(ϕ), we may assume that u, v ∈ D(ϕ). Then, by Stampacchia's lemma,

Ω |∇(u ∨ v)| p dx + Ω |∇(u ∧ v)| p dx = {u<v} (|∇v| p + |∇u| p ) dx + {u>v} (|∇u| p + |∇v| p ) dx + {u=v} (|∇v| p + |∇u| p ) dx = Ω (|∇u| p + |∇v| p ) dx (5.2) and ∂ Ω B(x, u ∨ v) dµ + ∂ Ω B(x, u ∧ v) dµ = {u<v} (B(x, v) + B(x, u)) dµ + {u>v} (B(x, u) + B(x, v)) dµ + {u=v} (B(x, u) + B(x, v)) dµ = ∂ Ω (B(x, u) + B(x, v)) dµ. (5.3)
Combining (5.2) and (5.3) we get that

ϕ (u ∧ v) + ϕ (u ∨ v) = ϕ(u) + ϕ(v).
Hence, by Theorem 2.2 (b), S is order-preserving.

We show that S D S. Since the semigroup S is order preserving, it suffices to show that ϕ D and ϕ satisfy (2.5). Let u, v ∈ L 2 (Ω), v ≥ 0. If u ∈ W 1,p 0 (Ω) or v ∈ D(ϕ), then the inequality (2.5) is trivially satisfied. So assume that u ∈ W 1,p 0 (Ω) ∩ L 2 (Ω) and v ∈ D(ϕ). Note that u = 0 p-quasi-everywhere on ∂ Ω. Proceeding similarly as above, we thus obtain that

ϕ D ((|u| ∧ v) • sgn(u)) + ϕ (|u| ∨ v) = = 1 p {|u|<v} (|∇u| p + |∇v| p ) dx + ∂ Ω∩{|u|<v} B(x, v) dµ + 1 p {|u|≥v} (|∇(v sgn(u))| p + |∇(|u|)| p ) dx + ∂ Ω∩{|u|≥v} B(x, |u|) dµ ≤ ϕ D (u) + ϕ(v).
Hence, S D S. Finally, we show that S S N . Since S N is order preserving, it suffices to show that ϕ and ϕ N satisfy (2.5). Indeed, let u Step 1. We claim ψ is lower semicontinuous, local, bi-monotone and for every u, v ∈ W 1,p (Ω) ∩ L 2 (Ω) one has (5.5) ψ(u ∨ u) + ψ(u ∧ v) ≤ ψ(u) + ψ(v).

∈ D(ϕ), v ∈ W 1,p (Ω) ∩ L 2 (Ω), v ≥ 0. Then it is clear that (|u| ∧ v) sgn(u), |u| ∨ v ∈ W 1,p (Ω) ∩ L 2 (Ω).
First, since ϕ is lower semicontinuous on W 1,p (Ω) ∩ L 2 (Ω) and since ϕ N is continuous on that same space, it follows that ψ is lower semicontinuous on W 1,p (Ω) ∩ L 2 (Ω).

Second, since ϕ N is local and since, by assumption, ϕ is local, it follows that ψ is local, too.

Third, it follows from the domination S S N and Theorem 2. Hence, the functional ψ is bi-monotone. Finally, let u, v ∈ W 1,p (Ω) ∩ L 2 (Ω). By Stampacchia's lemma,

ϕ N (u ∨ v) + ϕ N (u ∧ v) = ϕ N (u) + ϕ N (v).
Since S is order preserving, we also have

ϕ(u ∨ v) + ϕ(u ∧ v) ≤ ϕ(u) + ϕ(v).
The last two relations yield (5.5).

Step 2. Let ψ 1 , ψ 2 : (W 1,p (Ω) ∩ L 2 (Ω)) + → [0, +∞] be defined by ψ 1 (u) := ψ(u) and ψ 2 (u) := ψ(-u) (u ∈ (W 1,p (Ω) ∩ L 2 (Ω)) + ). By Step 1, the functionals ψ 1 and ψ 2 both satisfy the hypotheses in Theorem 2.3 (i) (with the space W 1,p (Ω) replaced by the space W 1,p (Ω) ∩ L 2 (Ω)). It follows from Theorem 2.3 together with Remark 2.4 (b) and (c) that there exist a finite, regular Borel measure µ on Ω which is absolutely continuous with respect to the p-capacity, and two functions B 1 , B 2 : Ω × R + → [0, +∞] satisfying hypothesis (H + ) in Theorem 2.3 such that (5.8) ψ i (u) = Ω B i (x, u(x)) dµ for every u ∈ D(ψ i ), i = 1, 2.

For every x ∈ Ω and every s ∈ R we set Theorem 2.1 is completely proved.

  with ∆ p u = div (|∇u| p-2 ∇u), is well-posed in L 2 (Ω) if it is complemented by Dirichlet boundary conditions u = 0 on (0, ∞) × ∂ Ω (perfectly conducting boundary) or by Neumann boundary conditions |∇u| p-2 ∂ u/∂ ν = 0 on (0, ∞) × ∂ Ω (perfectly isolating boundary).

  The p-capacity of a set A ⊆ R N is given by Cap p (A) := inf u p W 1,p (R N ) : u ∈ W 1,p (R N ) and there exists O ⊆ R N open, such that A ⊆ O and u ≥ 1 a.e. on O .

Theorem 2 . 1 .

 21 Let Ω ⊆ R N be a bounded domain with Lipschitz continuous boundary ∂ Ω, and let p ∈ (1, ∞). Let S be the semigroup generated by a convex, l.s.c. functional ϕ : L 2 (Ω) → [0, +∞], and let S D and S N be the semigroups generated by the Dirichlet-p-Laplace operator and the Neumann-p-Laplace operator, respectively. Then the following assertions are equivalent.

Remark 2 . 4 .

 24 (a) We point out that the representing measure µ and function B are not unique. For example, given a representing measure µ and a representing function B, and given any Borel measurable weight w : Ω → R + which is bounded from above and from below (away from zero), the weighted measure wdµ and the function B/w represent ψ, too. (b) On the other hand, the proof of Theorem 2.3 shows that for any pair ψ 1 , ψ 2 of lower semicontinuous, monotone, local functionals satisfying the inequality (2.8) one can find a common representing measure µ with supp[µ] ⊆ supp[ψ 1 ] ∪ supp[ψ 2 ] and two representing functions B 1 and B 2 satisfying the condition (H + ) of Theorem 2.3 such that

  Cap p,Ω (A) := inf u p W 1,p (Ω) : u ∈ W 1,p (Ω) and there exists O ⊆ R N open, such that A ⊆ O and u ≥ 1 a.e. on O ∩ Ω ;

3. 3 .

 3 The special case of quadratic forms (p = 2 and B quadratic). A particular situation occurs when all functionals in Theorem 2.1 are assumed to be quadratic. A functional ϕ : L 2 (Ω) → (-∞, +∞] is quadratic if there exists a symmetric, bilinear form (a, D(a)) u, u) if u ∈ D(a), +∞ else. Note that D(ϕ) = D(a) is a linear space in this case. A quadratic functional ϕ is convex and l.s.c. if and only if the associated form (a, D(a)) is positive and closed. In this case, A := ∂ ϕ is a linear, selfadjoint, nonnegative operator and the associated semigroup S is a C 0 -semigroup of linear, selfadjoint contractions. The functionals ϕ D and ϕ N are quadratic if and only if p = 2, and then ∂ ϕ D and ∂ ϕ N are the realizations of the Laplace operator with Dirichlet and Neumann boundary conditions, respectively. In the case when all functionals are quadratic, our Theorem 2.1 should be compared to Theorem 4.1 in Arendt & Warma [2], at least in the situation when Ω is a bounded domain with Lipschitz continuous boundary ∂ Ω (in [2], Ω is an arbitrary open set). Theorem 4.1 in [2] characterizes all symmetric local semigroups which are sandwiched between the semigroups S D and SN . However, there the generating bilinear form (a, D(a)) and the measure µ are both assumed to satisfy an additional regularity condition, namely that D(a) ∩ C( Ω) is dense in (D(a), • a ) and that µ is admissible [2, Definition 2.3]. Our Theorem 2.1

3. 5 . 3 . 6 .

 536 Locality of ϕ. Also the assumption that the functional ϕ is local is necessary. In fact, there are semigroups S generated by convex, l.s.c., and non-local functionals ϕ such that S D S S N . Clearly, such functionals can not be of the integral form as in Theorem 2.1 (ii). Examples of such non-local functionals exist even in the quadratic case, that is, when all semigroups are linear; see[START_REF] Arendt | Dirichlet and Neumann boundary conditions: What is in between?[END_REF] Example 4.5]. Relation between the lower semicontinuity of ϕ and properties of B and µ. Let Ω ⊆ R N be a bounded domain with Lipschitz continuous boundary ∂ Ω. Let µ be a finite, regular Borel measure on ∂ Ω (no further assumption on µ), and let B : ∂ Ω × R → [0, +∞] be a Borel function satisfying the hypothesis (H) from Theorem 2.1. Fix p ∈ (1, ∞), and consider the functional ϕ : L 2 (Ω) → [0, +∞] given by

  and lim n→∞ v n W 1,p (R N ) = 0. Now let u n := v n ∧ u. Then 0 ≤ uu n ≤ u everywhere on Ω, uu n = 0 on K, and lim n→∞ u n W 1,p (Ω) = 0.

Definition 4 . 1 .

 41 For each u ∈ D(ψ) we define a nonnegative set function µ u (•) on K by setting (4.1)

U n := {u < 1 /

 1 n} is p-quasi-open, that is, there exists a sequence (O j ) of open sets such that U n ∪ O j is open and lim j→∞ Cap p (O j ) = 0. Associated with (O j ) there exists a sequence (w j ) ⊆ W 1,p (R N ) such that 0 ≤ w j ≤ 1, w j = 1 on O j and lim j→∞ w j W 1,p (R N ) = 0. Note that lim j→∞ (u -1/n) + w j W 1,p (R N ) = 0 for every n ∈ N. Now, for n ∈ N, one may take u n = (u -1/n) + (1 -w j n ) with j n ∈ N large enough and one obtains the desired sequence.Let K ⊆ {u = 0} be a compact set. Since u n = 0 in a neighborhood of {u = 0}, it follows that K ⊆ Ω \ supp[u n ]. By definition of µ u n , since Ω \ supp[u n ] is relatively open (i.e. open with respect to the relative topology), and since ψ(0) = 0, one has µ u n (K) = 0. By Lemma 4.8, µ u (K) = lim n→∞ µ u n (K) = 0. The claim then follows from the inner regularity of µ u . Lemma 4.11. Let u, v ∈ D(ψ). Then µ u (G) ≤ µ v (G) for every Borel set G ⊆ {u ≤ v}. Proof. Step 1. We first prove the inequality µ u (G) ≤ µ v (G) for every Borel set G ⊆ {u < v}. By p-quasi-continuity of u and v, the set U := {u < v} is quasi-open. For every n ∈ N, let O n be an open set with Cap p (O n ) < 1/n and U n := U ∪O n is open. Associated with (O n ) there exists a sequence (w

Lemma 4 . 15 .

 415 The function B : Ω × R + → [0, +∞] defined above satisfies the hypothesis (H + ) of Theorem 2.3. Moreover, for every u ∈ D(ψ) one has B(•, u(•)) = B u (•) µ-almost everywhere on Ω.

  w n (x)<u(x) B w n (x) ≥ lim sup n→∞ B w α m+1 (n) (x) ≥ B λ m u (x) µalmost everywhere on {u > 0}. Since B λ m u → B u µ-almost everywhere on Ω by Lemma 4.14 (a), we thus obtain the remaining inequality B u (•) ≤ B(•, u(•)) µ-almost everywhere on {u > 0}. Proof of Theorem 2.3, (i) ⇒ (ii). Let (w n ) ⊆ D(ψ) be dense in D(ψ) and let µ be the Borel measure and B : Ω × R → [0, +∞] be the function defined above. It follows from Lemma 4.6, that supp[µ] ⊆ supp[ψ]. By Lemma 4.12, µ is absolutely continuous with respect to the p-capacity. By Lemma 4.15, the function B satisfies hypothesis (H + ) of Theorem 2.3. Now, by Lemma 4.15, for every u ∈ D(ψ) we have B(•, u(•)) = B u (•) µalmost everywhere on Ω. By the definition of B u this means

  Since

  ∂ Ω B(x, (|u| ∧ v) sgn(u)) dµ = {|u|≤v} B(x, u) dµ + {|u|>v} B(x, v sgn(u)) dµ ≤ {|u|≤v} B(x, u) dµ + {|u|>v} B(x, u) dµ = ∂ Ω B(x, u) dµ < ∞(where we have used the fact that B is bi-monotone), we have that (|u| ∧ v) sgn(u) ∈ D(ϕ). Now, calculating, we get thatϕ((|u| ∧ v) • sgn(u)) + ϕ N (|u| ∨ v) = = 1 p {|u|<v} (|∇u| p + |∇v| p ) dx + ∂ Ω∩{|u|<v} B(x, u) dµ + 1 p {|u|≥v} (|∇(v sgn(u))| p + |∇(|u|)| p ) dx + ∂ Ω∩{|u|≥v} B(x, v sgn(u)) dµ ≤ ϕ N (v) 1 p Ω |∇u| p dx + ∂ Ω∩{|u|<v} B(x, u) dµ + ∂ Ω∩{|u|≥v} B(x, v sgn(u)) dµ ≤ ϕ(u) + ϕ N (v),where we have again used the fact that B is bi-monotone. Hence, S is dominated by S N . The implication (ii)⇒(i) of Theorem 2.1 is completely proved. Proof of Theorem 2.1, (i) ⇒ (ii). Let S be the semigroup on L 2 (Ω) generated by a convex and lower semicontinuous functional ϕ : L 2 (Ω) → [0, +∞]. Assume that S is local, order preserving and S D S S N . Define the functional ψ : W 1,p (Ω) ∩ L 2 (Ω) → [0, +∞] by (5.4) ψ(u) = ϕ(u)ϕ N (u) if u ∈ D(ϕ), +∞ otherwise.

  2 (b) that D(ϕ) ⊆ D(ϕ N ) = W 1,p (Ω) ∩ L 2 (Ω),and(5.6)(ϕϕ N )(u) ≤ (ϕϕ N )(v) for all u, v ∈ D(ϕ) with 0 ≤ u ≤ v.The domination S S N and Theorem 2.2 (b) imply in addition that(5.7) (ϕϕ N )(v) ≤ (ϕϕ N )(u) for all u, v ∈ D(ϕ) with u ≤ v ≤ 0.

B 1

 1 (x, s) if s ≥ 0, B 2 (x, -s) if s < 0.It is readily seen that B satisfies the hypothesis (H). Since ψ is local, we obtain for every u ∈ D(ψ) = D(ϕ)ψ(u) = ψ(u +u -) = ψ(u + ) + ψ(-u -) = ψ 1 (u + ) + ψ 2 (u -),and hence, by (5.8) and (5.9),ψ(u) = Ω B 1 (x, u + (x)) dµ + Ω B 2 (x, u -(x)) dµ = Ω B(x, u) dµ, for every u ∈ D(ϕ). We have just shown that ϕ(u) = ϕ N (u) + ψ(u) = 1 p Ω |∇u| p dx + Ω B(x, u) dµ, for every u ∈ D(ϕ).Step 3. It follows from the domination S D S and Theorem 2.2 (b) thatW 1,p 0 (Ω) ∩ L 2 (Ω) = D(ϕ D ) ⊆ D(ϕ) and (ϕ Dϕ)(u) ≤ (ϕ Dϕ)(v) for all u, v ∈ W 1,p 0 (Ω) ∩ L 2 (Ω) with 0 ≤ u ≤ v.The domination S D S and Theorem 2.2 (b) imply in addition(ϕ Dϕ)(v) ≤ (ϕ Dϕ)(u) for all u, v ∈ W 1,p 0 (Ω) ∩ L 2 (Ω) with u ≤ v ≤ 0.These two inequalities together with the inequalities (5.6) and (5.7) imply that the functionals ϕ D , ϕ and ϕ N coincide on D(ϕ D ) = W 1,p 0 (Ω) ∩ L 2 (Ω), that is,ϕ D (u) = ϕ(u) = ϕ N (u) = 1 p Ω |∇u| p dx for every u ∈ D(ϕ D ) = W 1,p 0 (Ω) ∩ L 2 (Ω).It follows that the effective support of the functional ψ is a subset of ∂ Ω. In particular, by Theorem 2.3, supp[µ] ⊆ ∂ Ω. Hence, u) dµ for every u ∈ D(ϕ).

  Note that every local functional necessarily vanishes in 0. By abuse of language, we call a semigroup local if it is generated by a local functional.

The functionals ϕ D and ϕ N : L 2 (Ω) → [0, +∞] defined in (1.2) and (1.3) are convex, l.s.c. and local. Their effective domains D

  the semigroup S 2 is order preserving, then the semigroup S 1 is dominated by the semigroup S 2 , that is, S 1 S 2 , if and only if for every u

  1,p (Ω) ∩ L 2 (Ω). By convexity of ϕ, the set C is in addition convex. Hence, by Mazur's theorem, C is weakly closed in W 1,p (Ω) ∩ L 2 (Ω). Next, since B is nonnegative, the norm u → ∇u L p (Ω) + u L 2 (Ω) is bounded on C . This norm is equivalent to the canonical norm on W 1,p (Ω) ∩ L 2 (Ω): for p ≤ 2, this is always true, while for p > 2 we may use that Ω has a Lipschitz boundary and apply[START_REF] Maz'ya | Sobolev Spaces[END_REF] Section 1.1, Theorem]. Since the embedding W 1,p (Ω) ∩ L 2 (Ω) → L 2 (Ω) is continuous, we obtain that C is bounded and weakly closed, and hence closed, in L 2 (Ω). Since A was an arbitrary closed ball in L 2 (Ω), this shows that {ϕ ≤ c} is closed in L 2 (Ω).

Now let ρ i ∈ R(K i ) (i = 1, 2). Then, again by monotonicity and locality, ψ(uρ 1 ) + ψ(uρ 2 ) ≥ ψ(uρ 1 ρ 1 ) + ψ(uρ 2 ρ 2 ) = ψ(u(ρ 1 ρ 1 + ρ 2 ρ 2 ))

Since ρ i ∈ R(K i ) (i = 1, 2) were arbitrary, this implies µ u (K 1 ) + µ u (K 2 ) ≥ µ u (K 1 ∪ K 2 ).

Lemma 4.3 (Monotonicity). Let u ∈ D(ψ) and let K 1 , K 2 ⊆ Ω be two compact sets such that K 1 ⊆ K 2 . Then

Proof. This follows immediately from the definition of µ u and the inclusion R(K 1 ) ⊇ R(K 2 ).

Lemma 4.4. Let u ∈ D(ψ) and let K 1 , K 2 ⊆ Ω be two compact sets. Then

. Then, by assumption (2.8), Proof. First, the monotonicity of the set function µ u (Lemma 4.3) implies that lim m→∞ µ u (K m ) exists and lim m→∞ µ u (K m ) ≥ µ u (K). In order to prove the converse inequality, observe that for every δ , δ > 0 with δ > δ there exists m 0 ∈ N such that for every m ≥ m 0 one has

In particular, for every

Since this inequality holds for every ρ ∈ R(K), this proves µ u (K) ≥ lim m→∞ µ u (K m ). Step 2. Let now G be a Borel set in {u ≤ v and 0 < v}. Let (λ n ) ⊆ R + be a sequence such that λ n < 1 and lim n→∞ λ n = 1. Then, for every n one has {u ≤ v and 0 < v} ⊆ {λ n u < v}.

Hence, by

Step 1, for every n,

Clearly, λ n u ≤ u and lim n→∞ λ n u = u in W 1,p (Ω). Hence, by Lemma 4.8,

Step 3. Finally, let G be an arbitrary Borel set in {u ≤ v}. Then, by Step 2 and by Lemma 4.10,

which is the claim.

In the remainder of this section, fix a sequence (w n ) ⊆ D(ψ) such that {w n : n} is dense in D(ψ) (such a sequence exists since the space W 1,p (Ω) is separable). Then we define a measure µ on B by

It is clear that µ is a finite, regular Borel measure on Ω.

Lemma 4.12. The measure µ is absolutely continuous with respect to the p-capacity.

Proof. This follows directly from Lemma 4.9 and the definition of the measure µ.


Valentin Suder
email: valentin@suder.xyz

Valentin Suder Antiderivative Functions

Anne Canteaut

Gaëtan Leurent

Maria Naya

Antiderivative Functions over F 2 n

Keywords: Derivative functions, higher order derivative functions, antidifferentiation over F 2 n, antiderivative functions, linear structure, quadratic APN functions

Introduction

In symmetric cryptography, confusion in block and stream ciphers is usually performed by non-linear functions over finite fields called Substitution boxes (or S-boxes). In order to behave securely against most of the known attacks against cryptographic primitives using S-boxes, these functions must satisfy some security criteria. After the introduction of differential attacks [START_REF] Biham | Differential Cryptanalysis of DES-like Cryptosystems[END_REF], which is one of the most powerful known cryptanalysis, differential properties of functions over F 2 n have attracted a lot of attention from researchers not only in cryptography, but also in other related areas such as coding theory [START_REF] Carlet | Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems[END_REF] or finite geometry [START_REF] Edel | On quadratic APN functions and dimensional dual hyperovals[END_REF]. Indeed, the resistance of a function against differential attacks is closely related to the study of its discrete derivatives, especially of the size of their image sets [START_REF] Nyberg | Differentially uniform mappings for cryptography[END_REF]. The most resistant functions against this attack are called Almost Perfect Nonlinear (APN), and all of their derivatives have an image set of the largest possible size (see Definition 1 below). In terms of coding theory, these APN functions correspond to some binary codes having a minimal distance 5 [START_REF] Carlet | Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems[END_REF].

The generalization of differential cryptanalysis has brought new concepts and attacks linked to the derivative functions such as the impossible differential attacks [START_REF] Biham | Cryptanalysis of skipjack reduced to 31 rounds using impossible differentials[END_REF] or the boomerang attacks [START_REF] Wagner | The boomerang attack[END_REF] among others. Conversely, tools related to derivatives and their properties (e.g. algebraic degree), like higher order derivatives [START_REF] Lai | Higher Order Derivatives and Differential Cryptanalysis[END_REF], are frequently used in cryptanalysis, higher order differential attacks [START_REF] Knudsen | Truncated and Higher Order Differentials[END_REF], cube attacks [START_REF] Dinur | Cube attacks on tweakable black box polynomials[END_REF], zero-sum distinguishers [START_REF] Boura | Zero-Sum Distinguishers for Iterated Permutations and Application to Keccak-f and Hamsi-256[END_REF]. Therefore, building functions that fulfill differential criteria has become an important issue in discrete mathematics.

There already exist some efficient techniques to find APN functions, as the so-called switching method [START_REF] Edel | A new almost perfect nonlinear function which is not quadratic[END_REF]. Moreover, there are also techniques that aim at finding quadratic APN functions experimentally [START_REF] Weng | On Quadratic Almost Perfect Nonlinear Functions and Their Related Algebraic Object[END_REF][START_REF] Yu | A matrix approach for constructing quadratic APN functions[END_REF]. However, up to our knowledge, the idea of constructing such functions from the derivatives themselves has never been explored before. Indeed, the goal of the switching method, when applied to APN functions, is somehow to transform some Boolean component functions of an APN function in such a way that all the derivatives of these Boolean functions keep their differential properties.

In this paper, we introduce a method able to recover a function from a set of derivatives in given directions, with possibly some differential properties. In other words, we realize discrete antidifferentiation of functions over F 2 n .

In a recent work [START_REF] Xiong | Some results on the differential functions over finite fields[END_REF], Xiong et al. gave a criterion to decide whether or not a function is a derivative in a given direction, how many such derivatives exist, and what could be one of the function's antiderivative. However, their results were not pushed far enough, and many problems still arise. Indeed, in this paper, we revisit this work with another approach which allows us to not only discuss the higher order derivatives and their set, but also the functions that are canceled when differentiated in given directions and several other properties. In fact, higher order derivatives play an important role in the reconstruction of an antiderivative function. Most notably, we give a necessary and sufficient condition for some derivatives to be compatible, that is to be integrated in a function over F 2 n . We also provide an efficient algorithm that realizes this antidifferentiation. Then we introduce a new function equivalence over F 2 n , that we call differential equivalence, and which is distinct from CCZ-equivalence [START_REF] Carlet | Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems[END_REF]. Finally, we discuss the eventuality to recover APN functions from their derivatives. In particular, we show that the two presented recently [START_REF] Weng | On Quadratic Almost Perfect Nonlinear Functions and Their Related Algebraic Object[END_REF][START_REF] Yu | A matrix approach for constructing quadratic APN functions[END_REF] very similar techniques for finding quadratic APN functions, are quite surprisingly equivalent to the first natural application of antidifferentiation over F 2 n , which consists in gathering affine derivatives. This paper is thus organized as follows. After some basic definitions and notations (Section 2), in Section 3 we propose to see derivatives as linear applications described by matrices. By using techniques from linear algebra, we prove several properties concerning the set of derivatives and higher order derivatives. Then, in Section 4, we give the conditions that must be satisfied by the derivatives in order to be integrated into a discrete antidifference function, which we call antiderivative. We give also a simple algorithm that computes such antiderivatives. Finally, in Section 5, we talk about some applications and present the particular case of quadratic APN functions.

It is well known that any function F : F 2 n → F 2 n has a unique univariate polynomial representation:

F (x) = 2 n -1 i=0 f i x i ∈ F 2 n [x],
and its algebraic degree is given by deg(F ) = max{wt(i) | f i = 0}, where wt(i) is the binary Hamming weight of the integer i. In the rest of this paper, we will use both 'function' and 'polynomial' indifferently to describe such an element F . Definition 1. The discrete derivative or simply derivative of a function F :

F 2 n → F 2 n in a direction α ∈ F * 2 n is given by ∆ α F : F 2 n → F 2 n x → F (x) + F (x + α)
The differential uniformity of F , denoted by δ(F ), is defined as

δ(F ) = max α =0,β∈F 2 n {x | ∆ α F (x) = β} ≥ 2.
The differential uniformity of a function measures its resistance to differential cryptanalysis. APN functions are the ones that have the lowest possible differential uniformity, i.e. 2. A derivative that has exactly two pre-images for each element in its image set is called a 2-to-1 function. It is clear that every derivative of an APN function is 2-to-1.

Definition 2. Let F : F 2 n → F 2 n be a function and α, c ∈ F 2 n . We say that α is a c-linear structure of

F if ∆ α F (x) = F (x) + F (x + α) = c, for all x ∈ F 2 n .
Definition 3 ([12]). Let L ⊆ F 2 n be the vector space over F 2 spanned by the F 2 -linearly independent elements α 1 , . . . , α m ∈ F 2 n and let F :

F 2 n → F 2 n .
Then,

∆ L F (x) := ∆ α1,...,αm F (x) = ∆ α1 (∆ α2,...,αm F (x)) = c∈L F (x + c).
Remark that there is a one-to-one correspondence between functions from F 2 n into F 2 n and the vector space F 2 n 2 n . Definition 4. We denote by ϕ the bijective function that maps any function over F 2 n to the vector of its coefficients. More precisely, let F :

F 2 n → F 2 n such that F (x) = 2 n -1 i=0 f i x i , then ϕ(F ) =      f 0 f 1 . . . f 2 n -1      ∈ F 2 n 2 n .
For the sake of simplicity, we will denote ϕ(F ) by -→ F .

Thus, we have for all x ∈ F 2 n , F (x) = (0, x, x 2 , x 3 , . . . , x 2 n -1 ) • -→ F . Since our approach is mainly based on linear algebra, we also introduce the essential background on this topic. For more details, we invite the readers to refer to [START_REF] Paul | Finite-Dimensional Vector Spaces[END_REF].

Definition 5. The image and kernel of a linear application, given by its matrix M , between two vector spaces A and B, are denoted by Im(M ) and by ker(M ) respectively and are defined as follows:

Im(M ) = {y ∈ B | M •x = y, ∀x ∈ A} and ker(M ) = {x ∈ A | M •x = 0 B },
where 0 B denotes the null vector of the vector space B. Both Im(M ) and ker(M ) are vector spaces. The rank of a matrix M is defined as rank(M ) = dim(Im(M )).

Definition 6. Let M be a matrix defining an endomorphism of a vector space A and let A be a subspace of A:

1. A is said invariant under the action of M if x ∈ A implies M x ∈ A . 2. A + v = {a + v | a ∈ A }, with v ∈ A, is an affine space (or coset) of A.
3 Derivatives and higher order derivatives

A matrix point of view

First of all, note that any positive integer a can be uniquely written as a = i≥0 a i 2 i where a i 's lie in F 2 . We define the following relation: Definition 7. Let a and b be two positive integers. We will say that a is covered (resp. strictly covered) by b, and denote a b

(resp. a ≺ b), if {i | a i = 0} ⊆ {j | b j = 0} (resp. {i | a i = 0} ⊂ {j | b j = 0}).
We give a more precise expression of the derivative of a function

F (x) = i f i x i ∈ F 2 n [x], in a direction α ∈ F * 2 n .
For all x ∈ F 2 n , we have:

∆ α F (x) = F (x) + F (x + α) = 2 n -1 i=0 f i x i + 2 n -1 i=0 f i (x + α) i = 2 n -1 i=0 f i x i + 2 n -1 i=0 f i j, j i x j α i-j = 2 n -1 i=0 f i x i + 2 n -1 i=0 j, j i f i x j α i-j = 2 n -1 i=0 f i x i + 2 n -1 j=0 x j i, i j f i α i-j = 2 n -1 j=0 x j i, i j f i α i-j .
We can thus write ∆ α F (x) =

2 n -1 j=0 δ j (α)x j , where δ j (α) = i j f i α i-j . No- tice that each coefficient of ∆ α F (x) ∈ F 2 n [x], δ j (α)
, can be written as a vectorial product between coefficients f i 's and α i-j with i j. Then, from Definition 4, we have

---→ ∆ α F =      δ 0 (α) δ 1 (α)
. . .

δ 2 n -1 (α)      = M (α) • -→ F = M (α) •      f 0 f 1 . . . f 2 n -1      ,
where the matrix M (α) is defined as

M (α) = a i,j 0≤i,j≤2 n -1 , a i,j = α i-j if i j 0 otherwise .
We can now define the differentiation function over F 2 n 2 n , which gives the coefficients of the derivative of

F in a direction α ∈ F 2 n : -→ ∆ α : F 2 n 2 n → F 2 n 2 n -→ F → M (α) • -→ F
We can adapt some well known facts about derivatives and higher order derivatives [START_REF] Lai | Higher Order Derivatives and Differential Cryptanalysis[END_REF] to our matrix point of view:

Proposition 1. Let α 1 , . . . , α m ∈ F *
2 n , and F : F 2 n → F 2 n be any function:

1. --------→ ∆ α1,...,αm F = M (α 1 ) . . . M (α m ) • -→ F ; 2. Matrices M (α i )'s commute, that is ∆ αi,αj F (x) = ∆ αj ,αi F (x), for any 1 ≤ i, j ≤ m. 3. If α m is F 2 -linearly dependent from α 1 , . . . , α m-1 , then Π 1≤i≤m M (α i ) = 0.
In particular, M (α i ) is nilpotent of order 2, that is

M (α i ) 2 = 0 or ∆ αi,αi F (x) = 0. Example 1. Let n = 3 and α ∈ F * 2 n . M (α) =             . α α 2 α 3 α 4 α
            Block construction of M (α). Let α ∈ F * 2 n .
It is possible to construct the matrix M (α) by following the recursive method:

M 1 (α) = 0 α 0 0 , M i (α) = M i-1 (α) α 2 i-1 (Id i-1 +M i-1 (α)) 0 M i-1 (α) ,
where Id j is the identity matrix of size 2 j . Then M (α) := M n (α). We can notice here that even thought the size of the matrix M (α) grows exponentially with n, it is very sparse. Furthermore, since it has a nice block structure as well as several symmetries, practical computation with matrices M (α), even if n is quite large, is possible.

Characterization of derivative functions

Theorem 1. Let α ∈ F * 2 n . The rank of the matrix M (α) is rank(M (α)) = 2 n-1 .
Moreover, the kernel of the linear application defined by M (α) has dimension 2 n-1 and is generated by the matrix K(α):

K(α) = Id n-1 α 2 n-1 -1 M n-1 (α)
.

Sketch of Proof. We can verify easily that M (α)K(α) = 0. It implies that the vector space generated by the matrix K(α) is included in ker(M (α)) and dim(ker(M (α))) ≥ 2 n-1 . However, since M (α) = (m i,j ) with m i,j = α j-i if j i and 0 otherwise, M (α) is an upper triangular matrix and m i,i+1 = α for every even integer i and 0 otherwise. That is, M (α) has exactly 2 n-1 elements different from zero on the upper diagonal. Thus, rank(M (α)) ≥ 2 n-1 . We conclude by using the rank-nullity 1 theorem: dim(Ker(α)) + rank(M (α)) = 2 n .

In order to ease the notations, we will use Ker(α) to denote the vector space ker(M (α)), its basis matrix is then K(α).

Corollary 1.

Im(M (α)) = Ker(α).

Sketch of Proof.

From Proposition 1 (3.), we know that Im(M (α)) ⊂ Ker(α).

We conclude with Theorem 1, since dim(Im(M (α))) = dim(Ker(α)) = 2 n-1 .

This last corollary can be translated into the following:

Corollary 2. Let F : F 2 n → F 2 n and α ∈ F 2 n . Then, it exists a function G : F 2 n → F 2 n such that ∆ α G(x) = F (x) if and only if α is a 0-linear structure of F .
Note that, this characterization of derivative functions was already given in another way in [START_REF] Xiong | Some results on the differential functions over finite fields[END_REF]Theorem 1]. Nevertheless, the structure of the matrix K(α), permits to find and compute them efficiently even if n is large.

Higher order derivatives

Before discussing the general case of higher order derivatives, we first start with second order derivatives. This helps to understand better the general result that follows.

Lemma 1. Let α, β ∈ F * 2 n be distinct. Then,

Im (M (α)M (β)) = Ker(α) ∩ Ker(β) and (1)
ker (M (α)M (β)) = Ker(α) + Ker(β). (2)
Proof. We have that

Im(M (α)M (β)) = {M (α) • x | x ∈ Im(M (β))} = Im(M (α) |Im(M (β)) ).
Since M (α) and M (β) commute, we have that Im(M (β)) is invariant (see Definition 6 or [10, Section 39]) under the action of M (α). It means that we may ignore the fact that M (α) is defined outside Im(M (β)) and may consider M (α) as a linear transformation of Im(β). Thus, from Corollary 1, we have that

Im(M (α) |Im(M (β)) ) = ker(M (α) |Im(M (β)) ) = Ker(α) ∩ Im(M (α)) = Ker(α) ∩ Ker(β).
We prove Eq. ( 2). We know from the rank-nullity theorem and Eq. ( 1) that

dim(ker(M (α)M (β))) + rank(M (α)M (β)) = 2 n dim(ker(M (α)M (β))) + dim(Ker(α) ∩ Ker(β)) = 2 n .
Moreover, 2 n = dim(Ker(α)) + dim(Ker(β)). In other words, dim(ker(M (α)M (β))) = dim(Ker(α)) + dim(Ker(β)) -dim(Ker(α) ∩ Ker(β))

= dim(Ker(α) + Ker(β)).

We conclude this proof by noticing the following natural inclusion:

Ker(α) + Ker(β) ⊆ ker(M (α)M (β)).
We extend Lemma 1 to more than two distinct elements α, β in the next theorem.

Theorem 2. Let α 1 , . . . , α m ∈ F * 2 n be F 2 -linearly. Then, 1. the set of all m-order derivatives with respect to {α i } 1≤i≤m is

Im( -------→ ∆ α1, ..., αm ) = Im( 1≤i≤m M (α i )) = 1≤i≤m Ker(α i ). (3)
Furthermore, dim 1≤i≤m Ker(α i ) = 2 n-m .

2. the set of all functions such that their m-order derivative, with respect to

{α i } 1≤i≤m , cancel is ker( -------→ ∆ α1, ..., αm ) = ker( 1≤i≤m M (α i )) = 1≤i≤m Ker(α i ). (4)
Furthermore, dim 1≤i≤m Ker(α i ) = 2 n -2 n-m . Sketch of Proof.

1. Eq. ( 3) can be proven by induction on m with Eq. 1 as initialization and by noticing that matrices M (α i ) commute. We can prove the dimension by induction on m with the rank-nullity theorem and the fact that, for all distinct 1 ≤ i, j ≤ m:

dim(ker(M (α i )M (α j ))) = dim(Ker(α j )) + dim(Ker(α i ) ∩ Im(M (α j ))). 2. Let -→ v = m i=1 -→ v i with -→ v i ∈ Ker(α i ), that is -→ v ∈ m i=1 Ker(α i ). Then, ( m i=1 M (α i )) • -→ v = 0, and thus m i=1 Ker(α i ) ⊆ ker ( m i=1 M (α i )). From Eq. (3), we know that dim(ker( i≤i≤m M (α i ))) = 2 n -dim(Im( 1≤i≤m M (α i ))) = 2 n -2 n-m .
Now, we notice that with the vector spaces Ker(α i ), the intersection is distributive over the sum. Indeed, since Im(M (α i )) = Ker(α i ), from Eq. ( 1) and the fact that M (α i )'s commute, we have naturally, for

β ∈ F * 2 n , M (β)   1≤i≤m M (α i )   = 1≤i≤m (M (α i )M (β)) ⇒ Ker(β) ∩   1≤i≤m Ker(α i )   = 1≤i≤m (Ker(α i ) ∩ Ker(β)) .
Thus, we can apply the inclusion-exclusion principle (see [START_REF] Allenby | How To Count: An Introduction To Combinatorics[END_REF]Chapter 4]) to these particular vector spaces. Hence we obtain that dim

  1≤i≤m Ker(α i )   = 1≤k≤m (-1) k+1 m k 2 n-k .
The proof is completed by showing that the last equality is equal to 2 n -2 n-m , with the help of an induction on m.

Theorem 2 brings necessary and sufficient conditions to formalize higher order derivatives as in Corollary 2 for order 1. Most notably, we can now extract the following results:

Corollary 3. Let F : F 2 n → F 2 n , 1. if ∆ α1,...,αm F (x) = 0 then there are functions F 1 , . . . , F m : F 2 n → F 2 n , with ∆ αi F i (x) = 0, such that F (x) = F 1 (x)+• • •+F m (x) (the converse is obviously true). 2. if F possess some 0-linear structure in α 1 , . . . , α m F 2 -linearly independent, m < n, that is ∆ αi F (x) = 0, 1 ≤ i ≤ m, then, there exists a function G : F 2 n → F 2 n such that ∆ α1,...,αm G(x) = F (x) (the converse is obviously true).

Antidifferentiation

In this section, we introduce necessary and sufficient conditions as well as an algorithm to recover the antiderivative functions. From the sections above, we can first deduce the following proposition.

Proposition 2. Let F, G : F 2 n → F 2 n . They share the same derivatives in directions given by a subspace V of F 2 n if and only if every elements of V is a 0-linear structure of F + G.

Indeed, we have previously seen2 that:

∆ v F (x) = ∆ v G(x), ∀v ∈ V ⇔ -→ F + -→ G ∈ v∈V Ker(v). (5)
This results in a new function equivalence over F 2 n : Definition 8 (Differential equivalence). Let F, G : F 2 n → F 2 n . The functions F and G are said differentially equivalent with respect to a subspace V ⊆ F 2 n , denoted by

F ∼ V G, if ∆ v F (x) = ∆ v G(x), for all v ∈ V.
From Eq. ( 5), we have that the equivalence class, we call differential coset, of the function F : F 2 n → F 2 n with respect to the subspace V ⊆ F 2 n is the affine space ∩ v∈V Ker(v) + -→ F . In other words, a function G : F 2 n → F 2 n shares the same derivatives as F in directions given by V if and only if it consists in the sum of some higher order derivative function over V (see Eq. ( 3)) and F itself:

F ∼ V G ⇔ -→ G ∈ ∩ v∈V Ker(v) + -→ F . Furthermore, we know that the (algebraic) degree of a function h : F 2 n → F 2 n such that -→ h ∈ ∩ v∈V Ker(v) could not exceed n -dim(V )
. Indeed, we know that the algebraic degree of a function strictly decrease with its differentiation [12, Proposition 2]. Then, there is F :

F 2 n → F 2 n such that ∆ V F (x) = h(x)
and we have:

n ≥ deg(F ) > deg(∆ v F (x)) > • • • > deg(∆ V F (x)) ⇒ n -dim(V ) ≥ deg(h(x)).
In other words, the lower the dimension of the subspace, the lower the algebraic degree of the functions we can add in order to stay in the same differential coset. It is also clear that, in general, the differential equivalence does not preserve differential uniformity and is thus distinct from CCZ-equivalence [START_REF] Carlet | Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems[END_REF]. We can thus deduce the following proposition:

Proposition 3. Let F : F 2 n → F 2 n be a function and let V ⊂ F 2 n . Then, -if deg(F ) ≥ dim(V ), for all G : F 2 n → F 2 n such that F ∼ V G, we have deg(G) = deg(F ). -if dim(V ) ≥ n -1, for all G : F 2 n → F 2 n such that F ∼ V G, we have δ(F ) = δ(G).
We are now able to introduce the consistency theorem as well as an algorithm which computes the coefficients of antiderivative functions from consistent derivatives. By consistent, we mean derivatives that satisfy conditions in order to be integrated together, that is which are the derivative of a same function.

Theorem 3. Let α 1 , . . . , α m ∈ F * 2 n be F 2 -linearly independent elements and let f 1 , . . . , f m : F 2 n → F 2 n be such that -→ f i ∈ Ker(α i ), 1 ≤ i ≤ m. Then, there is at least one function F : F 2 n → F 2 n satisfying ∆ αi F (x) = f i (x), for all 1 ≤ i ≤ m, (6)
if and only if

∆ αj f i (x) = ∆ αi f j (x), for all 1 ≤ i, j ≤ m. (7)
Proof. Assume ∆ αi F (x) = f i (x), 1 ≤ i ≤ m, then ∆ αj f i (x) = ∆ αi,αj F (x) = ∆ αi f j (x).
Reciprocally, for all i, since -→ f i ∈ Ker(α i ), there is F i : F 2 n → F 2 n such that ∆ αi F i (x) = f i (x). Thus, from Eq. ( 7) and Theorem 2: ∆ αi,αj F j (x) = ∆ αj ,αi F i (x) ⇔ ∆ αi,αj (F i + F j )(x) = 0 ⇔ ----→ F i + F j ∈ Ker(α i ) + Ker(α j ).

Hence, there are

-→ G i ∈ Ker(α i ) and -→ G j ∈ Ker(α j ) such that ----→ F i + F j = -----→ G i + G j .
Moreover, for all 1 ≤ i ≤ m, we can choose G i to be the same each time F i is involved in a function. Thus, all the functions F i + G i , 1 ≤ i ≤ m, are equal and we can so denote by

F = F i + G i . Finally, we have ∆ αi F (x) = ∆ αi (F i + G i )(x) = ∆ αi F i (x) = f i for all 1 ≤ i ≤ m.
From Theorem 3, and due to the fact that both matrices M (α) and K(α), α ∈ F * 2 n are easy to implement efficiently, we end up with Algorithm 1. Notice that this algorithm only uses tools from linear algebra to compute solutions x of linear systems M x = y (we denote x = M \y). Moreover, the involved matrices are easy to handle. Note also that from Proposition 2, we know that the function in output is not unique, but the last matrix M , at the end of the computation, generates the subspace ∩ i Ker(α i ), so we can recover any differential equivalent functions.

Algorithm 1 Antidifferentiation over

F 2 n . function Antiderivative({(fi, αi) | 1 ≤ i ≤ m} verifying conditions in Th. 3) 1: M ← K(α1); -→ sol ← 0 F 2 n 2 n ; -→ F1 ← M (α1)\ -→ f1 ; 2: for i = 2 to i ≤ m do 3: -→ Fi ← M (αi)\ -→ fi ; 4: -→ sol ← -→ sol + M • (M (αi)M ) \ M (αi) • ---------→ F1 + Fi + sol ;
5: M ← M κ; κ: generating matrix of ker(M (αi)M ) 6: return sol + F0

Applications

In this section, we discuss the first natural application which consists in recovering quadratic APN functions from 2-to-1 affine functions. We briefly discuss the main idea of this method and show its equivalence to the previous works [START_REF] Weng | On Quadratic Almost Perfect Nonlinear Functions and Their Related Algebraic Object[END_REF][START_REF] Yu | A matrix approach for constructing quadratic APN functions[END_REF]. We believe also that our point of view permits a better understanding of these works.

From the previous sections (or [16, Proposition 1]), we know that every affine function,

L(x) = + i c i x 2 i ∈ F 2 n [x]
, is a derivative function if and only if L is not bijective. We can adapt Theorem 3 to fit affine functions and to find an easier condition since the differential of an affine function is constant:

Corollary 4. Let α 1 , . . . , α m ∈ F * 2 n be F 2 -linearly independent elements and let L 1 , . . . , L m : F 2 n → F 2 n be affine functions such that L i (α i ) = 0, 1 ≤ i ≤ m. Then there is at least one (quadratic) function F : F 2 n → F 2 n such that ∆ αi F (x) = L i (x), for all 1 ≤ i ≤ m, if and only if L i (α j ) + L i (0) = L j (α i ) + L j (0), for all 1 ≤ i, j ≤ m.
Note that a function is quadratic if and only if all of its derivatives are at most affines [START_REF] Berger | Priveate communication[END_REF].

Let α 1 , . . . , α n ∈ F * 2 n be a base of F 2 n over F 2 and consider the following matrix:

B = (β i,j ) 1≤i,j≤n =      β 1,1 β 1,2 β 1,3 . . . β 1,n β 2,1 β 2,2 . . . . . . . . . . . . β n,1 . . . β n,n      . (8)
If we assume that each row of this matrix is a base of a subspace of F 2 n , then there exist non-bijective affine functions L i : F 2 n → F 2 n satisfying (without loss of generality)

β i,j = L i (α j ) + L i (0), for all 1 ≤ i, j ≤ m.
Remark that in this case, each β i,i should be zero. Then, from each row of the matrix B in (8), we can recover the coefficients of the affine functions L i 's by a simple linear operation. Moreover, if we choose our β i,j wisely such that they satisfy the condition of Eq. ( 7) (i.e. the matrix B is symmetric), then there is a unique (quadratic) function (up to a constant, due to the differential equivalence) F : F 2 n → F 2 n , which we can easily recover with Algorithm 1, such that ∆ αi F (x) = L i (x), for all 1 ≤ i ≤ n.

Finally, if all non-zero elements of each row of B are F 2 -linearly independent, it induces that the image set of each affine function L i is an hyperplane (subspace of dimension 2 n-1 ), that is each L i is 2-to-1. Furthermore, if any linear combination of the rows of matrix B spans again an hyperplane, the antiderivative F must be APN.

This kind of matrices B we just described, are exactly the kind of matrices used in [START_REF] Weng | On Quadratic Almost Perfect Nonlinear Functions and Their Related Algebraic Object[END_REF] and [START_REF] Yu | A matrix approach for constructing quadratic APN functions[END_REF] and have permitted to discover hundreds of new quadratic APN functions. However, the method to justify them and to recover the function F is slightly different. Indeed, our algorithm to recover the antiderivative F is more general, since it would work theoretically with non-affine 2-to-1 compatible derivatives. We would thus end up with non-quadratic APN functions.

As another application of the tools presented in the sections above, we can look for functions differentially equivalent to APN functions with respect to a subspace of dimension ≤ n-3 (so that the functions to add in order to remain in the differential coset are quadratic). We performed a quick search on small fields (5 ≤ n ≤ 9) trying to find new APN functions from non-quadratic APN power functions with respect to a subspace of dimension n -3. Although exhaustive search in this differential cosets is expensive, after hours of computation, we did not find yet other APN functions in this differential coset. This leads us to express the following conjecture: Conjecture 1. By fixing 2 n-3 derivatives, in directions given by a subspace of F 2 n , of a non-quadratic APN power functions, there is no other APN functions in its differential coset.

Conclusion

In this paper, we provided a different point of view on derivatives and higher order derivatives over F 2 n . In the light of the above, concerning the possibility to recover APN functions, we shown that researchers should keep the focus on finding and studying class of 2-to-1 functions. Indeed, prior to this work, not much was known about what we could have done with 2-to-1 derivatives. The more we will know about these functions, the easier it will be to combine them (with respect to conditions of Theorem 3) and thus to recover APN functions using antidifferentiation over F 2 n .

On another hand, the differential equivalence permits a new classification of functions over F 2 n that will very probably lead to new constructions of functions with prescribed differential properties over F 2 n . Moreover, the matrix representation of derivatives we introduced in this paper should ease the future experiments.

Example 2 .1

 2 Let n = 3 and α ∈ F 2 n . The vector space Ker(α) is spanned by the columns of K(α): . . . . . . 1 . . α 8 . . . . . 1 . α 9 . . . . . . 1 α 10 α 9 α 8 See for instance [10, Section50].

Remark that Ker(α) ∩ Ker(β) ⊂ Ker(α + β) ⇐ ∆ α+β (∆ α,β F )(x) = 0 from Eq. (1).