Arnaud Debussche

AND Sylvain De Moor

Martina Hofmanov

A REGULARITY RESULT FOR QUASILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

We consider a non degenerate quasilinear parabolic stochastic partial differential equation with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Hölder continuous in time and possesses spatial regularity. Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problem whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs.

Introduction

In this paper, we are interested in the regularity of weak solutions of non degenerate quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise. Let D ⊂ R N be a bounded domain with smooth boundary, let T > 0 and set D T = (0, T ) × D, S T = (0, T ] × ∂D. We study the following problem (1.1)      du = div(B(u)) dt + div (A(u)∇u) dt + F (u) dt + H(u) dW, in D T , u = 0, on S T ,

u(0) = u 0 , in D.
where W is a cylindrical Wiener process on some Hilbert space K and H is a mapping with values in the space of the γ-radonifying operators from K to certain Sobolev spaces. The diffusion matrix A is assumed to be smooth and uniformly elliptic and the initial condition u 0 is random in general. The precise description of the problem setting will be given in the next section.

It is a well known fact in the field of PDEs and SPDEs that many equations do not, in general, have classical or strong solutions and can be solved only in some weaker sense. Unlike deterministic problems, in the case of stochastic equations we can only ask whether the solution is smooth in the Date: February 25, 2015. space variable since the time regularity is limited by the regularity of the stochastic integral. Thus, the aim of the present work is to determine conditions on coefficients and initial data under which there exists a spatially smooth solution to (1.1).

Such a regularity result is fundamental and interesting by itself. Equations of the form (1.1) appear in many sciences. Regularity of solutions is an important property when one wants to study qualitative behaviour. It is also a preliminary step when studying numerical approximations and implies strong uniqueness for (1.1). Our original motivation is that such models arise as limits of random kinetic equations (see [START_REF] Debussche | Diffusion limit, kinetic equations, stochastic partial differential equations, perturbed test functions[END_REF]).

The issue of existence of a classical solution to deterministic parabolic problems is well understood, among the main references stands the extensive book [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] which is mainly concerned with the solvability of initial-boundary value problems and the Cauchy problem to the basic linear and quasilinear second order PDEs of parabolic type. Special attention is paid to the connection between the smoothness of solutions and the smoothness of known data entering into the problem (initial condition and coefficients), nevertheless, due to the technical complexity of the proofs a direct generalization to the stochastic case is not obvious.

In the case of linear parabolic problems, let us mention the classical Schauder theory (see e.g. [START_REF] Lieberman | Second order parabolic differential equations[END_REF]) that provides a priori estimates relating the norms of solutions of initial-boundary value problems, namely the parabolic Hölder norms, to the norms of the known quantities in the problems. These results are usually employed in order to deal with quasilinear equations: the application of the Schauder fixed point theorem leads to the existence of a smooth solution under suitable hypotheses on the coefficients. In our proof, we make use of the Schauder theory as well, yet in an entirely different approach.

Regularity of parabolic problems in the stochastic setting was also studied in several works. In the previous work of the third author [START_REF] Hofmanová | Strong solutions of semilinear stochastic partial differential equations[END_REF], semilinear parabolic SPDEs (i.e. the diffusion matrix A being independent of the solution) were studied and a regularity result established by using semigroup arguments (see also [START_REF] Zhang | Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises[END_REF], [26]). In [START_REF] Denis | Maximum principle for quasilinear SPDE's on a bounded domain without regularity assumptions[END_REF], [START_REF] Denis | Maximum principle and comparison theorem for quasi-linear stochastic PDE's[END_REF], a maximum principle is obtained for an SPDE similar to (1.1) but with a more general diffusion H, it may depend on the gradient of u. In [START_REF] Gess | Strong solutions for stochastic partial differential equations of gradient type[END_REF], existence and uniqueness of strong solutions to SPDEs with drift given by the subdifferential of a quasi-convex function is proved. Hölder continuity of solutions to nonlinear parabolic systems under suitable structure conditions was proved in [START_REF] Beck | A regularity theorem for quasilinear parabolic systems under random perturbations[END_REF] by energy methods. Quasilinear stochastic porous media equations are studied in [START_REF] Barbu | Michael Existence of strong solutions for stochastic porous media equation under general monotonicity conditions[END_REF], [START_REF] Barbu | An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise[END_REF] and specific techniques for these equations are used. In comparison to this work, the quasilinear case considered in the present paper is more delicate and different techniques need to be applied.

The transposition of the deterministic method exposed in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] seems to be quite difficult. Fortunately, we may use a trick to avoid this. We use a very simple idea: a weak solution to (1.1) that satisfies a priori estimates is decomposed into two parts u = y + z where z is a solution to a linear parabolic SPDE with the same noise term as (1.1) and y solves a linear parabolic PDE with random coefficients. As a consequence, the problem of regularity of u is reduced to showing regularity of z and regularity of y which can be handled by known techniques for stochastic convolutions and deterministic PDEs. It is rather surprising that this classical idea used to treat semilinear equations can be applied also for quasilinear problems.

Let us explain this method more precisely. As the main difficulties come from the second order and stochastic terms, for simplicity of the introduction we assume B = F = 0. Let u be a weak solution to

(1.2) du = div (A(u)∇u) dt + H(u) dW, u(0) = u 0 ,
and let z be a solution to dz = ∆z dt + H(u) dW,

z(0) = 0.
Then z is given by the stochastic convolution with the semigroup generated by the Laplacian, denoted by (S(t)) t≥0 , i.e.

z(t) = t 0 S(t -s)H(u) dW (s)
and regularization properties are known. Setting y = u -z it follows immediately that y solves (1.3) ∂ t y = div(A(u)∇y) + div((A(u) -I)∇z),

y(0) = u 0 ,
which is a (pathwise) deterministic linear parabolic PDE. According to a priori estimates for (1.2), it holds for all p ∈ [2, ∞) and

u 0 ∈ L p (Ω × D): u ∈ L p (Ω; L ∞ (0, T ; L p (D))) ∩ L p (Ω; L 2 (0, T ; W 1,2 (D))),
and making use of the factorization method it is possible to show that z possesses enough regularity so that ∇z is a function with good integrability properties. Now, a classical result for deterministic linear parabolic PDEs with discontinuous coefficients (see [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]) yields Hölder continuity of y (in time and space) and consequently also Hölder continuity of u itself.

Having this in hand, the regularity of z can be increased to a level where the Schauder theory for linear parabolic PDEs with Hölder continuous coefficients applies to (1.3) (see [START_REF] Lieberman | Second order parabolic differential equations[END_REF]) and higher regularity of y is obtained. Repeating this approach then allows us to conclude that u is λ-Hölder continuous in time for all λ < 1/2 and possesses as much regularity in space as allowed by the regularity of the coefficients and the initial data. In this article, in order to avoid lengthy proofs and notations, we restrict to spatial regularity less than 5. Our method extends to higher regularity, see Remark 2.8.

The paper is organized as follows. In Section 2, we introduce the basic setting and state our regularity results, Theorem 2.6, Theorem 2.7. Section 3 gives preliminary results concerning the stochastic convolution and the smoothness of the solutions of linear parabolic equations. These are adapted from [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] and [START_REF] Lieberman | Second order parabolic differential equations[END_REF], we have to explicit the dependance of the constant with respect to some data and to treat low time regularity. The remainder of the paper is devoted to the proof of Theorem 2.6 and Theorem 2.7 that is divided into several parts. In Section 4, we establish our first regularity result, Theorem 2.6, that gives some Hölder continuity in time and space of a weak solution to (1.1). The regularity is then improved in the final Section 5 and Theorem 2.7 is proved.

2. Notations, hypotheses and the main result 2.1. Notations. In this paper, we adopt the following conventions. For r ∈ [1, ∞], the Lebesgue spaces L r (D) are denoted by L r and the corresponding norm by • r . In order to measure higher regularity of functions we make use of the Bessel potential spaces H a,r (D), a ∈ R and r ∈ (1, ∞).

In order to motivate the use of these spaces let us recall their basic properties (for a thorough exposition we refer the reader to the books of Triebel [START_REF] Triebel | Interpolation Theory, Function Spaces, Differential Operators, 2nd edi[END_REF], [START_REF] Triebel | Theory of Function Spaces II[END_REF]). In the case of R N the Bessel potential spaces are defined in terms of Fourier transform of tempered distributions:

let a ∈ R, r ∈ (1, ∞) then H a,r (R N ) = f ∈ S (R N ); f H a,r := F -1 (1 + |ξ| 2 ) a/2 Ff L r < ∞
and they belong to the Triebel-Lizorkin scale F a r,s (R N ) in the sense that H a,r (R N ) = F a r,2 (R N ). The Bessel potential spaces H a,r (R N ) behave well under the complex interpolation, i.e. for a 0 , a 1 ∈ R and r 0 , r

1 ∈ (1, ∞) it holds that (2.1) [H a 0 ,r 0 (R N ), H a 1 ,r 1 (R N )] θ = H a,r (R N ),
where θ ∈ (0, 1) and a = (1 -θ)a 0 + θa 1 , 1 r = 1-θ r 0 + θ r 1 , which makes them more suitable for studying regularity for linear elliptic and parabolic problems. Indeed, under the assumption of bounded imaginary powers of a positive operator A on a Banach space X, the domains of fractional powers of A are given by the complex interpolation as well: let 0

≤ α < β < ∞, θ ∈ (0, 1) then [D(A α ), D(A β )] θ = D(A (1-θ)α+θβ
). Furthermore, the expression (2.1) and the obvious identity H m,r (D) = W m,r (D) for m ∈ N 0 and r ≥ 1 suggest how the spaces H a,r (D) may be defined for a general domain D: if a ≥ 0 and m ∈ N such that a ≤ m < a+1 then we define

H a,r (D) := [W m,r (D), L r (D)] (m-a)/m .
If D is sufficiently regular then H a,r (D) coincides with the space of restrictions to D of functions in H a,r (R N ) and the Sobolev embedding theorem holds true. The spaces H a,r 0 (D), a ≥ 0, r ∈ (1, ∞), are then defined as the closure of C ∞ c (D) in H a,r (D). Note, that H a,r 0 (D) = H a,r (D) if a ≤ 1/r and H a,r 0 (D) is strictly contained in H a,r if a > 1/r. Besides, an interpolation result similar to (2.1) holds for these spaces as well [H a 0 ,r 0 0 (D), H a 1 ,r 1 0 (D)] θ = H a,r 0 (D). For notational simplicity, we denote the norm of H a,r by • a,r .

Remark 2.1. The spaces H a,r (D) are generally different from the Sobolev-Slobodeckij spaces W a,r (D) which belong to the Besov scale B a r,s (D) in the sense that W a,r (D) = B a r,r (D) if a > 0, a / ∈ N. Nevertheless, we have the following two relations which link the two scales of function spaces together

W a,r (D) = H a,r (D) if a ∈ N 0 , r ∈ [1, ∞) or a ≥ 0, r = 2,
and

H a+ε,r (D) → W a,r (D) → H a-ε,r (D) a ∈ R, r ∈ (1, ∞), ε > 0.
Below we use the Laplace operator with Dirichlet boundary conditions, denoted by ∆ D . Considered as an operator on L r , its domain is H 2,r 0 and it is the infinitesimal generator of an analytic semigroup denoted by S = (S(t)) t≥0 . Moreover, it follows from the above considerations that the domains of its fractional powers coincide with the Bessel potential spaces, that is D((-∆ D ) α ) = H 2α,r 0 , α ≥ 0. Therefore, one can build a fractional power scale (or a Sobolev tower, see [START_REF] Amann | Linear and Quasilinear Parabolic Problems[END_REF], [START_REF] Engel | Short course on operator semigroups[END_REF]) generated by (L r , -∆ D ) to get

(2.2) H 2α,r 0 , -∆ D,2α,r ; α ≥ 0 ,
where -∆ D,2α,r is the H 2α,r 0 -realization of -∆ D . Having this in hand, an important result [2, Theorem V.2.1.3] describes the behavior of the semigroup S in this scale. More precisely, the operator ∆ D,2α,r generates an analytic semigroup S 2α,r on H 2α,r 0 which is naturally obtained from S by restriction, i.e. S 2α,r (t) is the H 2α,r 0 -realization of S(t), t ≥ 0, and we have the following regularization property: for any δ > 0 and t > 0, S 2α,r (t) maps

H 2α,r 0 into H 2α+δ,r 0 with (2.3) S 2α,r (t) L(H 2α,r 0 ,H 2α+δ,r 0 ) ≤ C t δ/2 .
For notational simplicity of the sequel we do not directly specify the spaces where the operators ∆ D and S(t), t ≥ 0, are acting since this is always clear from the context.

Another important scale of function spaces which is used throughout the paper are the Hölder spaces. In particular, if X and Y are two Banach spaces and α ∈ (0, 1), C α (X; Y ) denotes the space of bounded Hölder continuous functions with values in Y equipped with the norm

f C α (X;Y ) = sup x∈X f (x) Y + sup x,x ∈X,x =x f (x) -f (x ) Y x -x α X .
In the sequel, we consider the spaces

C α (D) = C α (D; R), C α ([0, T ]; Y ) where Y = H a,r or Y = C β (D) and C α ([0, T ] × D) = C α ([0, T ] × D; R)...
Besides, we employ Hölder spaces with different regularity in time and space, i.e. C α,β ([0, T ] × D). For α ∈ (0, 1/2), β ∈ (0, 1), they are equipped with the norm

f C α,β = sup (t,x) |f (t, x)| + sup (t,x) =(s,y) |f (t, x) -f (s, y)| max{|t -s| α , |x -y| β } .
For larger indices α + l/2, β + k with α ∈ (0, 1/2), β ∈ (0, 1) and l, k non negative integers, the norm is defined by

f C α+l/2,β+k = 2r≤l,|γ|≤k,2r+|γ|≤max{l,k} ∂ r t ∂ γ f C α,β .
We have denoted by ∂ t the partial derivative with respect to the time variable t and, for a multi-index γ = (γ 1 , . . . , γ N ),

∂ γ = ∂ γ 1 1 . . . ∂ γ N
N where ∂ i is the partial derivative with respect to x i . For α = β/2, k = l, these are the classical Hölder spaces used to measure the regularity of solutions of parabolic problems. Here, we need these slightly more general spaces. In this work, we always use α = β/2. This is a very natural choice since in this case C α,β is precisely the Hölder space of order β with respect to the parabolic distance: d((t, x), (s, y)) = max{|t -s| 1/2 , |x -y|}.

Note that C β/2,β+1 ([0, T ] × D) = C (β+1)/2,β+1 ([0, T ] × D).
Indeed both spaces consist of bounded functions such that

f C β/2,β + |γ|=1 ∂ γ f C β/2,β is finite.
Given a domain D with a C β+k boundary, using local coordinates, we define classically the Hölder spaces on the boundary C α,β ([0, T ] × ∂D) (see [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], section 3, chapter II. In particular (3.19) and the paragraph above).

Clearly if v is a function in C α+l/2,β+k ([0, T ]×D), its restriction to [0, T ]× ∂D is in C α+l/2,β+k ([0, T ] × ∂D). But it can be much smoother, for instance if v is constant on the boundary.

Note that it holds

C α ([0, T ]; C β (D)) C α,β ([0, T ] × D
) and therefore we have to distinguish these two spaces. An example is given in [START_REF] Rabier | Vector-valued Morrey's embedding theorem and Hölder continuity in parabolic problems[END_REF]: D = [0, T ] and u(t, x) = (x + t) α , α ∈ (0, 1).

For k ∈ N 0 we denote by C k b = C k b (R) the space of continuous functions that have continuous bounded derivatives up to order k. Note that for k = 0 it is the space of continuous functions, not necessarily bounded.

In the whole article, C, C i , K, K i , κ, . . . denote constants. When they depend on some parameters of the problem, this is explicitly stated.

2.2.

Hypotheses. Let us now introduce the precise setting of (1.1). We work on a finite-time interval [0, T ], T > 0, and on a bounded domain D in R N with smooth boundary. We denote by D T the cylinder (0, T ) × D and by S T the lateral surface of D T , that is S T = (0, T ] × ∂D. Concerning the coefficients A, B, F, H, we only state here the basic assumptions that guarantee the existence of a weak solution and are valid throughout the paper. Further regularity hypotheses are necessary in order to obtain better regularity of the weak solution and will be specified later. We assume that the flux function

B = (B 1 , . . . , B N ) : R -→ R N
is continuous with linear growth. The diffusion matrix

A = (A ij ) N i,j=1
: R -→ R N ×N is supposed to be continuous, symmetric, positive definite and bounded. In particular, there exist constants ν, µ > 0 such that for all u ∈ R and ξ ∈ R N , (2.4)

ν|ξ| 2 ≤ A(u)ξ • ξ ≤ µ|ξ| 2 .
The drift coefficient F : R → R is continuous with linear growth.

Regarding the stochastic term, let (Ω, F , (F t ) t≥0 , P) be a stochastic basis with a complete, right-continuous filtration. The driving process W is a cylindrical Wiener process: W (t) = k≥1 β k (t)e k with (β k ) k≥1 being mutually independent real-valued standard Wiener processes relative to (F t ) t≥0 and (e k ) k≥1 a complete orthonormal system in a separable Hilbert space K.

For each u ∈ L 2 (D) we consider a mapping H(u) : K → L 2 (D) defined by H(u) e k = H k (•, u(•)).
In particular, we suppose that H k ∈ C(D × R) and the following linear growth condition holds true

(2.5) k≥1 |H k (x, ξ)| 2 ≤ C 1 + |ξ| 2 , ∀x ∈ D, ξ ∈ R.
This assumption implies in particular that H maps L 2 (D) to L 2 (K; L 2 (D)) where L 2 (K; L 2 (D)) denotes the collection of Hilbert-Schmidt operators from K to L 2 (D). Thus, given a predictable process u that belongs to L 2 (Ω; L 2 (0, T ; L 2 (D))), the stochastic integral t → t 0 H(u)dW is a well defined process taking values in L 2 (D) (see [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF]Chapter 4] for a thorough exposition).

Later on we are going to estimate the weak solution of (1.1) in certain Bessel potential spaces H a,r 0 with a ≥ 0 and r ∈ [2, ∞) and therefore we need to ensure the existence of the stochastic integral in (1.1) as an H a,r 0valued process. We recall that the Bessel potential spaces H a,r 0 with a ≥ 0 and r ∈ [2, ∞) belong to the class of 2-smooth Banach spaces since they are isomorphic to L r (0, 1) according to [23, Theorem 4.9.3] and hence they are well suited for the stochastic Itô integration (see [START_REF] Brzeźniak | On stochastic convolution in Banach spaces and applications[END_REF], [START_REF] Brzeźniak | Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process[END_REF] for the precise construction of the stochastic integral). So, let us denote by γ(K, X) the space of the γ-radonifying operators from K to a 2-smooth Banach space X. We recall that Ψ ∈ γ(K, X) if the series

k≥0 γ k Ψ(e k )
converges in L 2 ( Ω, X), for any sequence (γ k ) k≥0 of independent Gaussian real-valued random variables on a probability space ( Ω, F, P) and any orthonormal basis (e k ) k≥0 of K. Then, the space γ(K, X) is endowed with the norm

Ψ γ(K,X) := E k≥0 γ k Ψ(e k ) 2 X 1/2
(which does not depend on (γ k ) k≥0 , nor on (e k ) k≥0 ) and is a Banach space. Now, if a ≥ 0 and r ∈ [2, ∞) we denote by (H a,r ) the following hypothesis

H(u) γ(K,H a,r 0 ) ≤ C 1 + u H a,r 0 , a ∈ [0, 1], C 1 + u H a,r 0 + u a H 1,ar 0 , a > 1, (H a,r ) i.e. H maps H a,r 0 to γ(K, H a,r 0 ) provided a ∈ [0, 1] and it maps H a,r 0 ∩ H 1,ar 0 to γ(K, H a,r 0 ) provided a > 1.
The precise values of parameters a and r will be given later in each of our regularity results.

Remark 2.2. We point out that, thanks to the linear growth hypothesis (2.5) on the functions (H k ) k≥1 , one can easily verify that, for all r ∈ [2, ∞), the bound (H 0,r ) holds true.

In order to clarify the assumption (H a,r ), let us present the main examples we have in mind.

Example 2.3. Let W be a d-dimensional (F t )-Wiener process, that is W (t) = d k=1 W k (t) e k ,
where W k , k = 1, . . . , d, are independent standard (F t )-Wiener processes and (e k ) d k=1 is an orthonormal basis of K = R d . Then hypothesis (H a,r ) is satisfied for a ≥ 0, r ∈ [2, ∞) provided the functions H 1 , . . . , H d are sufficiently smooth and respect the boundary conditions in the following sense: if a > 1 r , then

∇ l x H k (x, 0) = 0, x ∈ ∂D, ∀k = 1, . . . , d, ∀l ∈ N 0 , l < a - 1 r ,
(for more details we refer the reader to [START_REF] Runst | Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations[END_REF]). Note that in this example it is necessary to restrict ourselves to the subspace

H a,r 0 ∩ H 1,ar 0 of H a,r 0 so that the corresponding Nemytskij operators u → H k (•, u(•)) take values in H a,r 0 . In fact, if 1 + 1/r ≤ a ≤ N/r, r ∈ (1, ∞), then only linear operators map H a,r
0 to itself (see [START_REF] Runst | Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations[END_REF]). Example 2.4. In the case of linear operator H we are able to deal with an infinite dimensional noise. Namely, let W be a (F t )-cylindrical Wiener process on K = L 2 (D), that is W (t) = k≥1 W k (t) e k , where W k , k ≥ 1, are independent standard (F t )-Wiener processes and (e k ) k≥1 an orthonormal basis of K. We assume that H is linear of the form H(u)e k := u Qe k , k ≥ 1, where Q denotes a linear operator from K to K. Then, one can verify that the hypothesis (H a,r ) is satisfied for a ≥ 0, r ∈ [2, ∞) provided we assume the following regularity property: k≥1 Qe k 2 a,∞ < ∞. We point out that, in this example, H maps H a,r 0 to γ(K, H a,r 0 ) for any a ≥ 0 and r ∈ [2, ∞).

As we are interested in proving the regularity up to the boundary for weak solutions of (1.1), it is necessary to impose certain compatibility conditions upon the initial data and the null Dirichlet boundary condition. To be more precise, since u 0 can be random in general, let us assume that u 0 : Ω → C(D) is measurable and u 0 = 0 on ∂D. Further integrability and regularity assumptions on u 0 will be specified later.

Note that other boundary conditions could be studied with similar arguments, see Remark 2.8 below.

Existence of weak solutions.

Let us only give a short comment here as existence is not our main concern and we will only make use of a priori estimates for parabolic equations of the form (1.1). In the recent work [START_REF] Debussche | Degenerate parabolic stochastic partial differential equations: Quasilinear case[END_REF], the authors gave a well-posedness result for degenerate parabolic SPDEs (with periodic boundary conditions) of the form

du = div(B(u)) dt + div(A(u)∇u) dt + H(u) dW, u(0) = u 0 ,
where the diffusion matrix was supposed to be positive semidefinite. One can easily verify that the Dirichlet boundary conditions and the drift term F (u) in (1.1) do not cause any additional difficulties in the existence part of the proofs and therefore the corresponding results in [START_REF] Debussche | Degenerate parabolic stochastic partial differential equations: Quasilinear case[END_REF], namely Section 4 (with the exception of Subsection 4.3) and Proposition 5.1, are still valid in the case of (1.1). In particular, we have the following.

Theorem 2.5. There exists ( Ω, F , ( Ft ), P), W , ũ which is a weak martingale solution to (1.1) and, for all p ∈ [2, ∞) and

u 0 ∈ L p ( Ω; L p ) ũ ∈ L p ( Ω; C([0, T ]; L 2 )) ∩ L p ( Ω; L ∞ (0, T ; L p )) ∩ L p ( Ω; L 2 (0, T ; W 1,2 )).
By a weak martingale solution we mean that it is weak in the PDE sense, that is, the equation (1.1) is satisfied in D ([0, T ) × D). It is also weak in probabilistic sense, that is, the probability space and the driving Wiener process are considered as unknown (see [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF]Chapter 8] for precise definition).

In the sequel, we assume the existence of a weak solution (in the PDE sense) on the original probability space (Ω, F , P) and show that it possesses regularity that depends on the regularity of coefficients and initial data. We point out that this assumption is taken without loss of generality since pathwise uniqueness can be proved once we have sufficient regularity in hand and hence existence of a pathwise solution can be then obtained by the method of Gyöngy and Krylov [START_REF] Gyöngy | Existence of strong solutions for Itô's stochastic equations via approximations[END_REF] (see also [START_REF] Debussche | Degenerate parabolic stochastic partial differential equations: Quasilinear case[END_REF]Subsection 4.3]).

2.4.

The main result. To conclude this section let us state our main results to be proved precisely.

Theorem 2.6. Let u be a weak solution to (1.1) such that, for all p ∈ [2, ∞),

u ∈ L 2 (Ω; C([0, T ]; L 2 )) ∩ L p (Ω; L ∞ (0, T ; L p )) ∩ L 2 (Ω; L 2 (0, T ; W 1,2 0 )). Assume that (i) u 0 ∈ L m (Ω; C ι (D))
for some ι > 0 and all m ∈ [2, ∞), and u 0 = 0 on ∂D a.s.

(ii) (H 1,2 ) is fulfilled.
Then there exists η > 0 such that, for all m ∈ [2, ∞), the weak solution u belongs to L m (Ω; C η/2,η (D T )).

Next result gives higher regularity. As in the deterministic case, some compatibility conditions are required on the boundary {0} × ∂D. To state these, we introduce the notation: u

(1) 0 = div (A(u 0 )∇u 0 ) + div(B(u 0 )) + F (u 0 ) and A 0 = ij A ij (0)∂ ij . Theorem 2.7. Let k = 1, 2, 3 or 4. Let u be a weak solution to (1.1) such that, for all p ∈ [2, ∞), u ∈ L 2 (Ω; C([0, T ]; L 2 )) ∩ L p (Ω; L ∞ (0, T ; L p )) ∩ L 2 (Ω; L 2 (0, T ; W 1,2 0 )).
Assume that

(i) u 0 ∈ L m (Ω; C k+ι (D)) for some ι > 0 and all m ∈ [2, ∞), (ii)
u 0 = 0 on ∂D a.s., and u

(1)

0 = 0 on ∂D a.s. if k = 2, 3, 4, and
2(A (0)∇u 0 ) • ∇u (1) 0 + B (0) • ∇u (1) 0 + A 0 u (1) 0 = 0 on ∂D if k = 4 . (iii) A, B ∈ C k b and F ∈ C k-1 b , (iv) (H a,r ) is fulfilled for all a < k + 1 and r ∈ [2, ∞).
Then for all λ ∈ (0, 1/2) and all m ∈ [2, ∞), the weak solution u belongs to L m (Ω; C λ,k+ι (D T )).

Remark 2.8. We could investigate higher regularity. Indeed, the tools required to prove Theorem 2.7 extend without difficulty. In fact, the proofs require different arguments for k = 1, 2, 3 and 4. But for k ≥ 5, the proof is exactly the same as for k = 4. This requires just a generalization of Theorem 3.5 below. Note that, as in the deterministic case, this requires stronger assumptions and compatibility conditions. The case of Neumann boundary conditions can be treated similarly.

Our result clearly extends to the case of coefficients A, B, F which may also depend on x under suitable regularity assumptions.

The case of periodic boundary conditions is much easier. Indeed, no compatibility conditions are required. Moreover, the proof of regularity higher than 3 is straightforward by differentiation of the equation. Remark 2.9. Note, that the condition m ∈ [2, ∞) in the assumption (i) of Theorem 2.6 and Theorem 2.7 can be weakened to m ∈ [2, m 0 ] for some sufficiently large m 0 . This would weaken the results accordingly. However, in the proof, we use big vallues of m and the precise value of m 0 is not of great interest. Therefore, we prefered to state our result in the form above. The solution to (3.1) is given by the stochastic convolution, that is

Z(t) = t 0 S(t -s)Ψ(s) dW s , t ∈ [0, T ].
In order to describe the connection between its regularity and the regularity of Ψ, we recall the following proposition.

Proposition 3.1. Let a ≥ 0 and r ∈ [2, ∞) and let Ψ be a progressively measurable process in L p (Ω;

L p (0, T ; γ(K, H a,r 0 ))). (i) Let p ∈ (2, ∞) and δ ∈ [0, 1 -2/p). Then, for any λ ∈ [0, 1/2 - 1/p -δ/2), Z ∈ L p (Ω; C λ (0, T ; H a+δ,r 0
)) and

E Z p C λ (0,T ;H a+δ,r 0 ) ≤ C E Ψ p L p (0,T ;γ(K,H a,r 0
)) . (ii) Let p ∈ [2, ∞) and δ ∈ (0, 1). Then Z ∈ L p (Ω; L p (0, T ; H a+δ,r 0 )) and

E Z p L p (0,T ;H a+δ,r 0 ) ≤ C E Ψ p L p (0,T ;γ(K,H a,r 0 )) .
Proof. Having established the behavior of the Dirichlet Laplacian and the corresponding semigroup along the fractional power scale (2.2), the proof of (i) is an application of the factorization method and can be found in [7, Corollary 3.5] whereas the point (ii) follows from the Burkholder-Davis-Gundy inequality and regularization properties (2.3) of the semigroup.

3.2.

Regularity results for deterministic parabolic PDE's. We now state classical regularity results from [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] and [START_REF] Lieberman | Second order parabolic differential equations[END_REF]. Since the notations are different and difficult to find in the books, we restate the results with ours and in the situation needed in the sequel. Moreover, the dependence on the coefficients and initial data is not always explicit in these books. We thus precise the bounds, this requires new proofs. Also, we give new results where the regularity is not measured in the classical parabolic scaling.

We first consider a linear parabolic PDE of the form

(3.2)      ∂ t v = div(a(t, x)∇v) + div g(t, x) + f (t, x), in D T , v = 0, on S T , v(0) = v 0 , in D,
and assume that there exist ν, µ > 0 such that ν|ξ| 2 ≤ a(t, x)ξ

• ξ ≤ µ|ξ| 2 , for all (t, x) ∈ D T , ξ ∈ R N .
This equation is precisely (1.9) with L defined in (1.1), from [19, Chapter III] with f i = g i , i = 1, . . . , N . Note that a is matrix valued. We have switched the notation for the unknown from u to v and for the initial data from ψ 0 to v 0 . Condition (1.2) in [19, Chapter III] is precisely the condition on a just above. Theorem 3.2. Assume that v 0 ∈ C β (D) for some β > 0 and v 0 = 0 on ∂D. There exists r 0 depending on N ; α ∈ (0, β] depending on N, ν, µ, r 0 , D and a constant K 1 depending on N, ν, µ, r 0 , D, α such that if (i) v is a weak solution to (3.2) which is continuous in time with values in L 2 and belongs to L 2 (0, T ; W 1,2 0 ), (ii) f ∈ L r 0 (D T ) and g ∈ L 2r 0 (D T ), then v ∈ C α/2,α (D T ) and the following estimate holds

v C α/2,α ≤ K 1 v 0 C α (D) + g L 2r 0 (D T ) + f L r 0 (D T ) . (3.3)
Proof. This result follows from [19, Theorem 10.1, Chapter III]. Note the compatibility condition v 0 = 0 on ∂D implies that the data on the parabolic boundary [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF].

Γ T = {0} × D ∪ S T is in C α/2,α (Γ T ), which is denoted by H α,α/2 (Γ T ) in
Numbers of equations and theorems below refer to [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]Chapter III].

We first have to check that v is in L ∞ ([0, T ] × D). This follows from Theorem 7.1..

We first note that it is always possible to find a number r 0 depending on N such that the couple (r, q) = (r 0 , r 0 ) satisfies (7.2) with some κ 1 ∈ (0, 1) if N ≥ 2 or κ 1 ∈ (0, 1 2 ) if N = 1. Hypothesis (7.1) with µ 1 depending on r 0 and N then follows. The crucial point is the estimate (7.14) then Theorem 6.1 from Chapter II applies and yields the maximum principle. Moreover, the dependence of the L ∞ -bound on the coefficients and the initial data can be seen from (6.2). We obtain

v L ∞ (D T ) ≤ C 1 + v 0 L ∞ 1 + g 2(1+ N 2κ 1 ) L 2r 0 (D T ) + f 1+ N 2κ 1 L r 0 (D T )
where the constant C does not depend on v 0 , g, f and depends on a only through its ellipticity constant ν. In other words we have proved that the linear mapping

L ∞ (D) × L r 0 (D T ) × L 2r 0 (D T ) → L ∞ (D T ) (v 0 , f, g) → v (3.4) is bounded hence continuous and it holds v L ∞ (D T ) ≤ C v 0 L ∞ + g L 2r 0 (D T ) + f L r 0 (D T ) . (3.5)
Since we assume zero Dirichlet boundary conditions we may now apply the second part of Theorem 10.1 and deduce that v is in C α/2,α (D T ). The proof is based on estimate (10.6) applied to cylinders Q(ρ, τ ) intersecting the parabolic boundary Γ T = {0} × D ∪ S T and to levels k ≥ max Q(ρ,τ )∩Γ T ±u and follows from Theorem 8.1 in Chapter II. Note that due to Remark 7.1 in Chapter II, the Hölder exponent α depends neither on the L ∞ -bound of v nor on the L r 0 -and L 2r 0 -norm of f and g, respectively, but may depend on the C β -norm of v 0 as can be seen from the proof of Theorem 8.1, Chapter II, namely from (8.6). Nevertheless, one can first take v 0 C β ≤ 1 and then argue by linearity of the solution map (3.4) to obtain the final estimate (3.3).

As the next step, we recall the Schauder estimate for equations in divergence form [START_REF] Lieberman | Second order parabolic differential equations[END_REF]Theorem 6.48].

Theorem 3.3. Let α ∈ (0, 1), assume that D has a C α+1 boundary and

(i) a, g ∈ C α/2,α (D T ), f ∈ L p (D T ) for some p ≥ N +2 1-α , (ii) v 0 ∈ C 1+α (D), v 0 = 0 on ∂D.
Then there exists a unique weak solution to (3.2). Moreover, there exists a constant K 2 depending on N, ν, µ, p, D, α such that

v C (1+α)/2,1+α ≤ K 2 P 1 (a α/2,α ) v 0 C 1+α (D) + g C α/2,α + f L p (D T ) , (3.6)
where P 1 is a polynomial and a α/2,α = a C α/2,α .

Proof. According to [START_REF] Lieberman | Second order parabolic differential equations[END_REF]Theorem 6.48] one obtains the required regularity of v. Note that our compatibility condition v 0 = 0 on ∂D implies that the boundary data are in C (1+α)/2,1+α on the parabolic boundary S T . This parabolic boundary is denoted by PΩ in [START_REF] Lieberman | Second order parabolic differential equations[END_REF] and this Hölder space by H 1+α . Thus the compatibility condition of [20, Theorem 6.48] is satisfied.

But the dependence on a α/2,α is not obvious. In the following we adapt the technique from [9, Theorem 3.1] and show that the dependence is indeed polynomial.

Take r ∈ (0, 1] to be fixed later. Let (B i (r/4)) i=1,...,n be a covering of D T by parabolic cylinders of radius r, i.e. balls of radius r with respect to the parabolic distance d((t, x), (s, y)) = max{|t -s| 1/2 , |x -y|}.

We denote their centers by (t i , x i ). Let

ϕ i ∈ C ∞ c (B i (r)) such that 0 ≤ ϕ i ≤ 1 and ϕ i ≡ 1 on B i (r/2).
It is possible to choose these functions such that

∇ϕ i C 0 (D T ) ≤ Cr -1 , ∂ t ϕ i C 0 (D T ) ≤ Cr -2 , ∇ϕ i C α/2,α ≤ Cr -1-α . Set a i = a(t i , x i ), v i = ϕ i v, v 0,i = ϕ i (0)v 0 . Then v i solves the following parabolic equation (3.7)          ∂ t v i -div(a i ∇v i ) = div[(a -a i )∇v i ] + ϕ i f + div(ϕ i g) -g∇ϕ i -a∇v∇ϕ i -div[av∇ϕ i ] + v∂ t ϕ i , in D T , v i = 0, on S T , v i (0) = v 0,i , in D.
This is an equation of the form

     ∂ t w -div(b∇w) = h 0 + div h, in D T , w = 0, on S T , w(0) = w 0 , in D,
where b is a constant matrix. According to [START_REF] Lieberman | Second order parabolic differential equations[END_REF]Theorem 6.48], it holds true

w C (1+α)/2,1+α ≤ C w 0 C 1+α + h C α/2,α + h 0 M 1,N +1+α (D T ) , (3.8)
where the constant depends only on N, ν, µ, D, α and M 1,N +1+α (D T ) denotes the Morrey space (corresponding to the parabolic distance). It is easy to see that the compatibility conditions are satisfied by (3.7).

By Hölder inequality it follows from the definition of the norm in

M 1,N +1+α (D T ) that if p ≥ N +2 1-α then L p (D T ) → M 1,N +1+α (D T ). Therefore w C (1+α)/2,1+α ≤ C 0 w 0 C 1+α + h C α/2,α + h 0 L p (D T ) (3.9)
with a constant depending only on N, ν, µ, D, α.

We apply this bound to (3.7) and obtain

v i C (1+α)/2,1+α ≤ C 0 ϕ i (0) C 1+α v 0 C 1+α + (a -a i )∇v i C α/2,α + f L p (D T ) + ϕ i C α/2,α g C α/2,α + ∇ϕ i C 0 (D T ) g L p (D T ) + a C 0 (D T ) ∇ϕ i C 0 (D T ) ∇v L p (D T ) + a C α/2,α v C α/2,α ∇ϕ i C α/2,α + v C 0 (D T ) ∂ t ϕ i C 0 (D T ) . (3
v i C 1/2,1 ≤ ε v i C (1+α)/2,1+α + C ε 1/α v i C 0 (D T ) .
Besides,

∇v i C 0 (D T ) ≤ v i C 1/2,1 ∇v i C 0 (D T ) + |∇v i | α/2,α ≤ v C (1+α)/2,1+α hence choosing ε = r α we obtain (a -a i )∇v i C α/2,α ≤ a -a i C 0 (B i (r)) ∇v i C 0 (D T ) + a -a i C 0 (B i (r)) |∇v i | α/2,α + a α/2,α ∇v i C 0 (D T ) ≤ a α/2,α r α ∇v i C 0 (D T ) + a α/2,α r α |∇v i | α/2,α + a α/2,α ∇v i C 0 (D T ) ≤ a α/2,α 2r α v i C (1+α)/2,1+α + Cr -1 v i C 0 (D T ) .
We set r = 4C 0 a α/2,α -1/α ∧ 1, where C 0 is the constant defined in (3.9), and deduce from (3.10)

v i C (1+α)/2,1+α ≤ C r -1-α v 0 C 1+α + a α/2,α r -1 v i C 0 (D T ) + f L p (D T ) + r -1 g C α/2,α + a α/2,α r -1 ∇v L p (D T ) + a α/2,α r -1-α v C α/2,α + r -2 v C 0 (D T ) .
Next, we estimate the second, the sixth and the seventh term on the right hand side by (3.3) and (3.5) and obtain

v i C (1+α)/2,1+α ≤ Cr -1-α ∇v L p (D T ) + C r -1-2α + r -2 v 0 C 1+α + f L p (D T ) + g C α/2,α . (3.11)
Since p ≥ 2 we also have the elementary interpolation inequality

∇v p L p (D T ) ≤ ∇v p-2 C 0 (D T ) ∇v 2 L 2 (D T )
as well as the basic energy estimate

(3.12) v L ∞ (0,T ;L 2 ) + ∇v L 2 (D T ) ≤ C v 0 L 2 + g L 2 (D T ) + f L 2 (D T )
which holds true with a constant that depends on ν but otherwise is independent of the coefficients of (3.2). We deduce

∇v p L p (D T ) ≤ C ∇v p-2 C 0 (D T ) v 0 C 1+α + f L p (D T ) + g C α/2,α 2
Therefore, Young's inequality yields

Cr -1-α ∇v L p (D T ) ≤ Cr -1-α ∇v p-2 p C 0 (D T ) v 0 C 1+α + f L p (D T ) + g C α/2,α 2 p ≤ Cr -p 2 (1+α) v 0 C 1+α + f L p (D T ) + g C α/2,α + 1 2 ∇v C 0 (D T ) .
Hence it follows from (3.11)

v i C (1+α)/2,1+α ≤ 1 2 ∇v C 0 (D T ) + C r -1-2α + r -2 + r -p 2 (1+α) v 0 C 1+α + f L p (D T ) + g C α/2,α .
(3.13)

Let (t 0 , x 0 ) ∈ D T such that |∇v(t 0 , x 0 )| = ∇v C 0 (D T )
and take i 0 such that (t 0 , x 0 ) ∈ B i 0 (r/4). Then ∇v(t 0 , x 0 ) = ∇v i 0 (t 0 , x 0 ). Therefore

∇v C 0 (D T ) ≤ ∇v i 0 C 0 (D T )
and consequently

∇v C 0 (D T ) ≤ C r -1-2α + r -2 + r -p 2 (1+α) v 0 C 1+α + f L p (D T ) + g C α/2,α . (3.14)
Plugging this back in (3.13) we obtain

v i C (1+α)/2,1+α ≤ C r -1-2α + r -2 + r -p 2 (1+α) v 0 C 1+α + f L p (D T ) + g C α/2,α .
In order to estimate

v (1+α)/2,1+α = v C 0 (D T ) + ∇v C 0 (D T ) + |∇v| α/2,α ,
we apply (3.5), (3.14) and the inequality

|∇v| α/2,α ≤ max max i |∇v i | α/2,α , 2 1+2α r -α ∇v C 0 (D T )
which is easily obtained by considering the points where the maximum in the definition of |∇v| α/2,α is reached. Indeed, if (t, x) and (s, y) are such points, then |∇v| α/2,α is bounded by 2 1+2α r -α ∇v C 0 (D T ) provided d((t, x), (s, x)) ≥ r 4 and by |∇v i | α/2,α otherwise, where i is such that (t, x), (s, y) ∈ B i (r/2). We deduce

v (1+α)/2,1+α ≤ C(r -1-3α + r -2-α + r -p+αp+2α 2 v 0 C 1+α + f L p (D T ) + g C α/2,α
which implies (3.6).

We now give higher regularity results for equation in non divergence form. We consider a linear parabolic PDE of the form

(3.15)            ∂ t v = N i,j=1 a ij (t, x)∂ i,j v + f (t, x), in D T , v = 0, on S T , v(0) = v 0 , in D,
and assume that the matrix a(t, x) is symmetric and that there exist ν, µ > 0 such that ν|ξ| 2 ≤ a(t, x)ξ • ξ ≤ µ|ξ| 2 , for all (t, x) ∈ D T , ξ ∈ R N . We use a similar approach to obtain regularity as for the proof of Theorem 3.3. In particular, this enables us to quantify the constant with respect to the parameters. We define A as the second order differential operator: Aw = N i,j=1 a i,j ∂ i,j w. We first recall low regularity in the parabolic scaling.

Theorem 3.4. Let α ∈ (0, 1), k = 0, 1 and assume that D has a C α+k+2 boundary. Consider the equation:

(3.16)      ∂ t v -Av = f, in D T , v = φ, on S T , v(0) = v 0 , in D.
Assume that

(i) f ∈ C (α+k)/2,α+k (D T ), a ∈ C (α+k)/2,α+k (D T ). (ii) v 0 ∈ C α+k+2 (D), (iii) φ ∈ C (α+k)/2+1,α+k+2 (S T ) (iv) v 0 = φ, ∂ t φ -Av 0 = f on {0} × ∂D.
Then there exists a unique solution v ∈ C (α+k)/2+1,α+k+2 (D T ) and

v C (α+k)/2+1,α+k+2 ≤ K 3 P 2 (A α+k ) f C (α+k)/2,α+k + v 0 C α+k+2 (D)
+ φ C (α+k)/2+1,α+k+2 (S T ) , (3.17

)
where K 3 depends only on µ, ν, D, α, P 2 is a polynomial and A α+k = a C (α+k)/2,α+k .

Proof. According to Theorem 5.2, chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], one obtains the required regularity of v. Indeed, H l,l/2 in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] corresponds to C l/2,l in our notations and (iv) is the compatibility condition of order one. But as in Theorem 3.3, the dependence on A α+k is not obvious, we proceed similarly.

Take r ∈ (0, 1] to be fixed. Let (B i (r/4)) i=1,...,n be a covering of D T by parabolic cylinders of radius r, i.e. balls of radius r with respect to the parabolic distance, and center (t i , x i ). Let

ϕ i ∈ C ∞ c (B i (r)) such that 0 ≤ ϕ i ≤ 1 and ϕ i ≡ 1 on B i (r/2), such that ∇ϕ i C 0 (D T ) ≤ Cr -1 , ∂ t ϕ i C 0 (D T ) ≤ Cr -2 , ∇ϕ i C α/2,α ≤ Cr -1-α , ∂ γ ϕ i C α/2,α ≤ Cr -2-α ,
for any multi-index γ of length |γ| = 2.

Set A i = A(t i , x i ), v i = ϕ i v, v 0,i = ϕ i (0)v 0 , φ i = ϕ i φ.
Then v i solves the following parabolic equation

(3.18)                  ∂ t v i -A i v i = v∂ t ϕ i + (A -A i )v i -vAϕ i -2 N l,m=1 a lm ∂ l v∂ m ϕ i + f ϕ i , in D T , v i = φ i , on S T , v i (0) = v 0,i , in D.
This equation is similar to (3.16) but the operator has constant coefficients.

It is easy to see that it satisfies the compatibility condition of order one. Thus, according to Theorem 5.2, chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], we have

v i C (α+k)/2+1,α+k+2 ≤ C v 0,i C α+k+2 (D) + φ i C (α+k)/2+1,α+k+2 (S T ) + v∂ t ϕ i + (A -A i )v i -vAϕ i -2 N l,m=1 a lm ∂ l v∂ m ϕ i + f ϕ i C (α+k)/2,α+k ,
where the constant depends only on N, ν, µ, D, α. To see this, one follows the proof of Theorem 5.2, Chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] for the case of constant coefficients. First, the equation is transformed by a local change of coordinates into a heat equation on the half space. In the case of zero initial data (see the problem (5.3)' in Chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]) one obtains the above estimate in Theorem 6.1, Chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]. Reduction of the problem to the problem with zero initial data is based on Theorem 4.3, Chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] and the method is explained in Section 8, Chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF].

The end of the proof is exactly as above for Theorem 3.3 and uses the maximum principle for parabolic equations in non divergence form, see e.g. [6, Theorem 8, Chapter 2].

We now study higher regularity. Since the application we have in mind is SPDEs, we do not have more than 1/2 regularity in time. Hence, to increase the spatial regularity, we cannot use the parabolic scaling anymore. Unfortunately, all the results in the literature are in the parabolic scaling. To avoid lengthy technical proofs, we only investigate the first step: spatial regularity of order k + α, k = 4, α < 1. Similar arguments can be used to get higher regularity. Theorem 3.5. Let α ∈ (0, 1), and assume that D has a C α+4 -boundary. Let v be the solution of

     ∂ t v -Av = f, in D T , v = 0, on S T , v(0) = v 0 , in D.
Assume that

(i) f ∈ C α/2,α+2 (D T ), f | S T ∈ C α/2+1,α+2 (S T ), a ∈ C α/2,α+2 (D T ), and a| S T ∈ C α/2+1,α+2 (S T ). (ii) v 0 ∈ C α+4 (D), (iii) v 0 = 0, Av 0 + f = 0, and ∂ t f + A 2 v 0 + Af + (∂ t A)v 0 = 0, on {0} × ∂D. then v ∈ C α/2+1,α+4 (D T ) and v C α/2+1,α+4 ≤ K 4 P 3 (A α+2 ) f C α/2,α+2 + f | S T C α/2+1,α+2 (S T ) + v 0 C α+4 (D) , (3.19)
where P 3 is a polynomial, A α+2 = a C α/2,α+2 + a| S T C α/2+1,α+2 (S T ) and K 4 depends only on α, ν, µ, D. Remark 3.6. It might seem strange that we need to assume more time regularity for f on the boundary S T . However, it is immediate that a solution in

C α/2+1,α+4 (D T ) satisfies, -Av = f on S T . Thus, f | S T ∈ C α/2+1,α+2 (S T ).
Concerning the compatibility conditions (iii), the first two are already in Theorem 3.4. Since the equation holds up to the boundary, we have -Av = f on S T . Also, the solution is sufficiently regular to differentiate it and get

∂ t (Av) -(∂ t A)v -A 2 v = Af so that on S T : -∂ t f -(∂ t A)v -A 2 v = Af
and for t = 0 we see that the third condition is also necessary.

Note that this third compatibility condition is exactly the classical second order condition required to get smoothness in the parabolic scaling (see [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], section 5, chapter IV. In particular (5.6) and below).

We set in the following result: R N,+ = {x ∈ R N : x N ≥ 0}, R N,++ = {x ∈ R N : x N > 0}. Note that its boundary is R N -1 . As usual, for x = (x 1 , . . . , x N ) ∈ R N , we set x = (x 1 , . . . , x N -1 ). Also, given a function g defined on R N,+ , we denote by g| x N =0 its restriction to {x N = 0}.

The proof of Theorem 3.5 follows the lines of the proof of Theorem 5.2, chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]. It is more complicated than the proof of Theorem 3.4 since we cannot base our argument on known results. We start with the case of the Laplace operator on the half space. The following Lemma provides the missing ingredient to reproduce the argument of [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF] which is based on inequalities (2.1), (2.2), (2.3) from Chapter IV. Inequality (3.20) is the generalization of these to indices that are not in the parabolic scaling. Lemma 3.7. Let α ∈ (0, 1) and v be the solution of

       ∂ t v -∆v = f, in R N,++ × (0, T ], v = 0, on R N -1 × (0, T ], v(0) = v 0 , in R N,++ . Assume that (i) f ∈ C α/2,α+2 (R N,+ × [0, T ]) and f | x N =0 ∈ C α/2+1,α+2 (R N -1 × [0, T ]), (ii) v 0 ∈ C α+4 (R N,+ ), (iii) v 0 = 0, -∆v 0 = f and ∂ t f + ∆ 2 v 0 + ∆f = 0 for x N = 0, t = 0. then v ∈ C α/2+1,α+4 (R N,+ × [0, T ]) and v C α/2+1,α+4 (R N,+ ×[0,T ]) ≤ K 5 f C α/2,α+2 (R N,+ ×[0,T ]) + f C α/2+1,α+2 (R N -1 ×[0,T ]) + v 0 C α+4 (R N,+ ) , (3.20)
where K 5 depends only on α and N .

Proof. We already know from Theorem 3.

4 that v ∈ C α/2+1,α+2 (R N,+ × [0, T ]). Let w = ∂ ij v, with i = N, j = N . It satisfies        ∂ t w -∆w = ∂ ij f, in R N,++ × (0, T ], w = 0, on R N -1 × (0, T ], w(0) = ∂ ij v 0 , in R N,++ .
Clearly, ∂ ij f and ∂ ij v 0 satisfy the assumption of Theorem 3.4. We deduce that

∂ ij v ∈ C α/2+1,α+2 (R N,+ × [0, T ]). It remains to prove that ∂ 2 n v ∈ C 1+α/2,α+2 (R N,+ × [0, T ]). We write w = ∂ 2 n v. Note that: w = ∂ t v -∆ x v -f.
Hence taking the restriction at x N = 0:

w| x N =0 = -f.
We deduce that w satisties:

       ∂ t w -∆w = ∂ 2 n f, in R N,++ × (0, T ], w = -f, on R N -1 × (0, T ],
w(0) = ∂ 2 n v 0 , in R N,++ . Our assumptions imply that ∂ 2 n f , ∂ 2 n v 0 and f | S T satisfy the assumptions of Theorem 3.4. We deduce that w ∈ C α/2+1,α+2 (R N,+ × [0, T ]) and this finishes the proof. Note that (3.20) follows from the estimate given on ∂ ij v and ∂ 2 n v by Theorem 3.4. Following the argument in section 6, chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], we generalize Lemma 3.7 to the case of a general elliptic operator with constant coefficient. Lemma 3.8. Let A = ij āij ∂ ij is a second order partial differential operator with constant coefficients such that ν|ξ| 2 ≤ āξ • ξ ≤ µ|ξ| 2 , for all ξ ∈ R N . Let α ∈ (0, 1) and v be the solution of

       ∂ t v -Av = f, in R N,++ × (0, T ], v = 0, on R N -1 × (0, T ], v(0) = v 0 , in R N,++ .
Assume that

(i) f ∈ C α/2,α+2 (R N,+ ×[0, T ]) and f | x N =0 ∈ C α/2+1,α+2 (R N -1 ×[0, T ]) (ii) v 0 ∈ C α+4 (R N,+ ), -Av 0 = f and ∂ t f + A 2 v 0 + Af = 0 for x N = 0, t = 0. then v ∈ C α/2+1,α+4 (D T ) and v C α/2+1,α+4 (R N,+ ×[0,T ]) ≤ K 6 f C α/2,α+2 (R N,+ ×[0,T ]) + f | x N =0 C α/2+1,α+2 (R N -1 ×[0,T ]) + v 0 C α+4 (R N,+ ) ,
where K 6 depends on α, N , µ and ν.

The proof of Theorem 3.5. Follows now exactly the proof given in section 7, chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]. It is quite long and we do not reproduce it here.

Note contrary to Theorem 3.4, we do not know that v has the required regularity. This has to be proved. As above, the proof consists in using a suitable covering of the domain D T into small sub-domains. On each of these domains intersecting the boundary, the equation is transformed by a local change of coordinate into an equation on the half space. This latter equation is approximated by the same equation with frozen coefficients, whose solution can be estimated thanks to Lemma 3.8. If the sub-domain does not intersect the boundary, an estimate is trivially obtained by differentiation of the equation. Putting together all these local inverses, we show that we obtain an approximate inverse. Than an elementary argument allows to conclude.

Note that contrary to [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], we work with non zero initial data (see the discussion after Theorem 5.3 for the definition of this notion). The price to pay is that we need to pay particular attention to the compatibility conditions but this does not cause any problem.

The polynomial dependence on A α+2 comes from similar computations as in the proof of Theorem 3.4. In the notation of section 7, chapter IV in [START_REF] Ladyzhenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], this comes from the choice of τ . It has to be chosen small enough so that T < 1/2 and the proof clearly shows that this condition can be written in terms of A α+2 and the local coordinate systems on ∂D.

4. First step in the regularity problem: proof of Theorem 2.6

In this section, we show the first step towards regularity of the weak solution u to (1.1). We consider the auxiliary problem (3.1) with Ψ = H(u), whose solution is given by the stochastic convolution This way, we have split u into two parts, i.e. y and z, that are much more convenient in order to study regularity.

Proof of Theorem 2.6.

Step 1: Regularity of z. According to the hypothesis, the weak solution u to (1.1) belongs to L 2 (Ω; L 2 (0, T ; H 1,2 0 )) so that, thanks to the hypothesis (H 1,2 ), we have that H(u) belongs to L 2 (Ω; L 2 (0, T ; γ(K, H 1,2 0 ))). As a result, with Proposition 3.1 -(ii) and the bound (H 1,2 ) we have that for any a ∈ (0, 2), z ∈ L 2 (Ω; L 2 (0, T ; H a,2 0 )) with

(4.3) E z 2 L 2 (0,T ;H a, 2
0 ) ≤ C 1 + E u 2 L 2 (0,T ;H 1,2 0 )
and by Proposition 3.1 -(i), for p > 2, (4.4)

E z p C([0,T ];L 2 ) ≤ C 1 + E u p L p (0,T ;L 2 ) .
Besides, since for all p ∈ [2, ∞), the weak solution u to (1.1) belongs to L p (Ω; L p (0, T ; L p )), we obtain, with the hypothesis (H 0,p ) (see Remark 2.2), that H(u) belongs to L p (Ω; L p (0, T ; γ(K, L p ))). As a consequence, with Proposition 3.1 -(ii) and the bound (H 0,p ) we have that for any b ∈ (0, 1), z ∈ L p (Ω; L p (0, T ; H b,p 0 )) with

E z p L p (0,T ;H b,p 0 ) ≤ C 1 + E u p L p (0,T ;L p ) .
We have proved that for any a ∈ (0, 2) and b ∈ (0, 1), we have z ∈ L 2 (Ω; L 2 (0, T ; H a,2 0 )) and z ∈ L p (Ω; L p (0, T ; H b,p 0 )). We can now interpolate to obtain that (see [START_REF] Amann | Compact embeddings of vector-valued Sobolev and Besov spaces[END_REF])

z ∈ L r (Ω; L r (0, T ; H c,r 0 )), where, for θ ∈ (0, 1),    1 r = θ 2 + 1 -θ p , c = aθ + b(1 -θ),
with the bound (4.5)

E z r L r (0,T ;H c,r 0 ) ≤ E z 2 L 2 (0,T ;H a,2 0 ) rθ/2 E z p L p (0,T ;H b,p 0 ) r(1-θ)/p < ∞.
Note that by choosing θ ∈ (0, 1) and p ∈ [2, ∞) appropriately, r can be arbitrary in [2, ∞). Furthermore, when θ ∈ (0, 1) is fixed, it is always possible to take (a, b) ∈ (0, 2) × (0, 1) such that c > 1. As a result, for all r ∈ [2, ∞), there exists c r > 1 such that z ∈ L r (Ω; L r (0, T ; H cr,r 0 )). This gives, for all r ∈ [2, ∞), ∇z ∈ L r (Ω; L r (0, T ; L r )), and, due to the boundedness of the mapping A, (A(u) -I)∇z ∈ L r (Ω; L r (0, T ; L r )), with, thanks to (4.5), (4.6) E (A(u) -I)∇z r L r (0,T ;L r ) ≤ CE z r L r (0,T ;H c,r 0 ) < ∞, where C > 0 depends on µ from (2.4). Note that, thanks to the linear growth property of the coefficients B and F , we obviously have, for all r ∈ [2, ∞),

(4.7) E B(u) r L r (0,T ;L r ) + E F (u) r L r (0,T ;L r ) ≤ C(1 + E u r L r (0,T ;L r ) ) < ∞.
Step 2: Regularity of y. We apply Theorem 3.2 with

a = A(u), g = B(u) + ((A(u) -I)∇z), f = F (u).
The assumptions are satisfied. Indeed (2.4) gives the required uniform ellipticity and boundedness of a. Then sublinear growth of B, F , Step 1 and (4.3) imply that assumption (ii) is satisfied. By the basic energy estimate for parabolic equations (3.12) we obtain that y ∈ L ∞ (0, T ; L 2 ) ∩ L 2 (0, T ; W 1,2 0 ) a.s. Besides, since u ∈ C([0, T ]; L 2 ) a.s. due to our assumptions and the same is valid for z due to (4.4), assumption (i) holds true as well. We may conclude

y C α/2,α ≤ K 1 u 0 C α (D) + B(u) + (A(u) -I)∇z L 2r 0 (D T ) + F (u) L r 0 (D T ) (4.8)
for α ≤ ι where α is given by Theorem 3.2. Therefore

E y m C α/2,α ≤ K 1 E u 0 m C α (D) + E B(u) + (A(u) -I)∇z m L 2r 0 (D T ) + E F (u) m L r 0 (D T ) . (4.9)
We now use (4.6)-(4.7) to deduce that the above right hand side is finite thanks to our assumptions on u.

Step 3: Hölder regularity of z. In order to complete the proof it is necessary to improve the regularity of z. Recall that for all m ∈ [2, ∞), the solution u to (1.1) belongs to L m (Ω; L m (0, T ; L m )) and that H(u) belongs to L m (Ω; L m (0, T ; γ(K, L m ))). We now apply Proposition 3.1 -(i) and (H 0,m ) to obtain, since H a,r

0 ⊂ H a,r , that for m ∈ (2, ∞), δ ∈ (0, 1 -2/m) and γ ∈ [0, 1/2 -1/m -δ/2), z ∈ L m (Ω; C γ ([0, T ]; H δ,m )) with E z m C γ ([0,T ];H δ,m ) ≤ C 1 + E u m L m (0,T ;L m
) . Note that we can choose δ and γ to be independent of m. For instance, let us suppose in the sequel that m ≥ 3; then δ = 1/6 and γ = 1/12 satisfies the conditions above for any m ≥ 3. Furthermore, from now on, we also suppose that m ≥ 7N := m 0 . This implies that m ≥ 3 and δm > N , so that the following Sobolev embedding holds true

H δ,m → C λ , λ := δ -N/m 0 .
We conclude that, for all m ≥ m 0 , (4.10)

E z m C γ ([0,T ];C λ ) ≤ C 1 + E u m L m (0,T ;L m ) < ∞.
Note that for m ∈ [2, m 0 ), we can write with the Hölder inequality

(4.11) E z m C γ ([0,T ];C λ ) ≤ E z m 0 C γ ([0,T ];C λ ) m/m 0 < ∞.
Step 4: Conclusion. Finally, we set η := min(α, 2γ, λ) > 0 and we recall that u = y + z so that the conclusion follows from (4.9), (4.10), (4.11) due to the fact that C η/2 ([0, T ]; C η (D)) ⊂ C η/2,η ([0, T ] × D).

5.

Increasing the regularity: proof of Theorem 2.7

In this final section, we complete the proof of Theorem 2.7. Having Theorem 2.6 in hand, it is now possible to significantly increase the regularity of u using the results given in Subsection 3.2.

We treat differently the cases k = 1, k = 2 and k = 3, 4.

5.1. The case k = 1. The proof is divided in two parts: we first increase the regularity in space and then in time.

Step 1: Regularity of z. First, we improve the regularity of z that was defined in (4.1). According to Theorem 2.6, there exists η > 0 such that for all m ∈ [2, ∞), u ∈ L m (Ω; C η/2,η (D T )). In particular, since u satisfies Dirichlet boundary conditions, this implies that u ∈ L m (Ω; L m (0, T ; H κ,m 0 )) provided κ < η. With (H κ,m ), we deduce that H(u) ∈ L m (Ω; L m (0, T ; γ(K, H κ,m 0 ))). An application of Proposition 3.

1 yields that z ∈ L m (Ω; C γ ([0, T ]; H κ+δ,m 0 )) for every m ∈ (2, ∞) with E z m C γ ([0,T ];H κ+δ,m 0 ) ≤ C 1 + E u m L m (0,T ;H κ,m 0 ) ,
where δ ∈ (0, 1 -2/m) and γ ∈ [0, 1/2 -1/m -δ/2). In the sequel, we assume that m ≥ (N + 4)/κ := m 0 . Then δ = 1 -3/m 0 and γ = 1/(4m 0 ) satisfies the conditions above uniformly in m ≥ m 0 . Furthermore, observe that (κ + δ)m > κm ≥ κm 0 ≥ N so that the following Sobolev embedding holds true H κ+δ,m → C σ , σ = κ + δ -N/m 0 .

Besides, by definition of m 0 , σ = κ + 1 -(N + 3)/m 0 > 1.

We deduce that there exists γ > 0, σ > 1 such that for all m ≥ m 0 , z ∈ L m (Ω; C γ ([0, T ]; C σ (D))) with ) .

Step 2: Regularity of y. Next, we improve the regularity of y that is given by (4.2).

As a consequence of Theorem 2.6, (4.7) and (5.1), we obtain due to the assumptions upon A, B and F that, for all m ∈ [2, ∞) where κ < α and λ ∈ (0, 1/2 -1/m). Therefore for any λ ∈ (0, 1/2) there exists m 0 large enough so that (5.3) holds true for any m ≥ m 0 and the Sobolev embedding then implies that z ∈ L m (Ω; C λ ([0, T ]; C 1+β (D))) for β < κ. Since we already have (5.2) we deduce that u ∈ L m (Ω; C λ,1+α (D T )) for any λ ∈ (0, 1/2) and m ∈ N.

Step 4: Conclusion. It is now possible to reproduce the 3 steps above. In step 1, we can now take κ < 1 + α. Then )) for κ < 1 + ι, γ < 1/2 -1/m -δ/2, δ < 1 -2/m and any m ∈ N. In particular, it is possible to take κ + δ -1/m > 2 and we deduce that z and its first and second derivatives vanish on ∂D.

5.2.

The case k = 2. Step 1: Regularity of z. Again we first increase the regularity of z. We know that for any λ ∈ (0, 1/2), there exists β > 0 such that for all m ∈ [2, ∞), u ∈ L m (Ω; C λ,1+β (D T )). We deduce

E z m C γ ([0,T ];H 1+κ+δ,m ) ≤ C 1 + E u m L m (0,T ;H 1+κ,m ) + E u (1+κ)m
L (1+κ)m (0,T ;H 1,(1+κ)m ) < ∞, where κ < β, δ ∈ (0, 1 -2/m) and γ ∈ [0, 1/2 -1/m -δ/2). By a similar reasoning as above we obtain due to the Sobolev embedding that there exist γ > 0 and σ > 2 such that z ∈ L m (Ω; C γ ([0, T ]; C σ (D))) for m ∈ [2, ∞).

Step 2: Regularity of y. In order to improve the space regularity of y we use Theorem 3.4. In particular, we set Finally, we iterate the argument as in step 4 of the case k = 1 and this completes the proof.

a ij = A ij (u), , f = ij A ij (

3 . Preliminaries 3 . 1 .

 331 Regularity of the stochastic convolution. Our proof of Theorem 2.6 and Theorem 2.7 is based on a regularity result that concerns mild solutions to linear SPDEs of the form (3.1) dZ = ∆ D Z dt + Ψ(t) dW t , Z(0) = 0, where ∆ D is the Laplacian on D with null Dirichlet boundary conditions acting on various Bessel potential spaces.

( 4 ∂

 4 .1) z(t) = t 0 S(t -s)H(u s ) dW s , t ∈ [0, T ].Next, we define the process y := u -z. It follows immediately that y solves the following linear parabolic PDE with random coefficients (t y = div (A(u)∇y) + div(B(u)) + F (u) + div ((A(u) -I)∇z) , in D T , y = 0, on S T , y(0) = u 0 , in D.

(5. 1 )

 1 E z m C γ ([0,T ];C σ (D)) ≤ C 1 + E u m L m (0,T ;H κ,m0

A)≤ C 1 +)

 1 (u) ∈ L m (Ω; C α/2,α (D T ), B(u) + (A(u) -I)∇z ∈ L m (Ω; C α/2,α (D T ),F (u) ∈ L m (Ω; L m (0, T ; L m )), u 0 ∈ L m (Ω; C 1+α (D)),where α := min(ι, η, σ -1, 2γ) > 0. Thus the hypotheses of Theorem 3.3 are fulfilled and we obtain the following (pathwise) estimatey C (1+α)/2,1+α ≤ K 2 P 1 (a α/2,α ) u 0 C 1+α (D) + B(u) + (A(u) -I)∇z C α/2,α + F (u) L r (0,T ;L r ) , where r ∈ [2, ∞) is large enough. We conclude that, for all m ∈ [2, ∞), (5.2) y ∈ L m (Ω; C (1+α)/2,1+α (D T ))which together with (5.1) yields u ∈ L m (Ω; C γ,1+α (D T )).Step 3: Time regularity. Having in hand the improved regularity of u, we consider again the stochastic convolution z, repeat the approach from the first step of this proof and obtain due to Theorem 2.6 (with δ = 0) and (H 1+κ,m ) E z m C λ ([0,T ];H 1+κ,m 0 E u m L m (0,T ;H 1+κ,m 0

Remark 5 . 1 .

 51 E z m C γ ([0,T ];C σ ) ≤ C 1 + E u m L m (0,T ;H κ,m 0 ) . with σ = κ + δ -N/m, γ < 1/2 -1/m -δ/2, δ < 1 -2/m. Let ε < (1 -ι)/2, m ≥ max(4, 2N )1/ε and δ = (1 -α)/2, then we can take σ -1 = 2γ = (1 -α)/2 -ε.Thus in step 2, we can reproduce the argument with α replaced by α 1 = min(ι, (1 -α)/2 -ε) and conclude u ∈ L m (Ω; C λ,1+α 1 (D T )) for any λ ∈ (0, 1/2) and m ∈ N. If α 1 < ι, we reproduce this argument and define recursively α n+1 = min(ι, (1-α n )/2-ε). In a finite number of step, we have α n = ι.The proof is complete for k = 1. Note that reproducing step 1, we can finally prove that z ∈ L m (Ω; C γ ([0, T ]; H κ+δ,m 0

)

  u)∂ i u∂ j y + div B(u) + (A(u) -I)∇z + F (u).According to the results above, we havea ij , f ∈ L m (Ω; C α/2,α (D T )), u 0 ∈ L m (Ω; C 2+α (D)), (5.4) for some α ∈ (0, σ -2] and all m ∈ [2, ∞) provided A, B ∈ C 2 b , F ∈ C 1b . The compatibility conditions (iv) are: u 0 = 0 andij A ij (u)∂ ij u = ij A ij (u)∂ i u∂ j y + div B(u) + (A(u) -I)∇z + F (u) = 0, on {0} × ∂D.The first one is clearly satisfied. For the second one, we use Remark 5.1 to rewrite it as:ij A ij (0)∂ ij u 0 = ij A ij (0)∂ i u 0 ∂ j u 0 + div B(u 0 ) + F (u 0 ) = 0,rearranging the terms this is exactly u (1) 0 . Thus, Theorem 3.4 applies and we deduce y ∈ L m (Ω;C α/2+1,α+2 (D T )), hence u ∈ L m (Ω; C γ,α+2 (D T )).Step 3: Time regularity. Finally, we improve the time regularity of u by considering the stochastic convolution again as in Subsection 5.1. We obtain that for any λ ∈ (0, 1/2) there exists m 0 large enough so thatE z m C λ ([0,T ];H 2+κ,m ) ≤ C 1 + E u m L m (0,T ;H 2+κ,m0 any m ≥ m 0 and the Sobolev embedding then implies that z ∈ L m (Ω; C λ ([0, T ]; C 2+β (D))) for β < κ.

5.3.

The case k = 3, 4. The case k = 3 is treated exactly as above using Theorem 3.4 except that (5.4) is replaced by (5.5) for some α ∈ (0, σ -3] where σ > 3.

For k = 4, we argue similarly but apply Theorem 3.5. The only thing we need to check is that f is smooth in time on the boundary and the validity of the third compatibility condition in (iii). Recall that f = ij A ij (u)∂ i u∂ j y+ div B(u) + (A(u) -I)∇z + F (u). Since z and its derivatives up to order 4 vanish on ∂D, we have

In the proof for k = 3, we get in Step 2 that y ∈ L m (Ω;

Thus, on the boundary

In particular, it is constant in time and the last term of the compatibility condition vanishes. Moreover on {0}×∂D, ∂ t u = u

(1) 0 = 0, we deduce:

Also, still on {0} × ∂D,

Regrouping terms, we obtain:

0 . Therefore our assumption in Theorem 2.7 implies the compatibility condition.

[26] X. Zhang


PhD Stéphanie Baulac
email: stephanie.baulac@upmc.fr

PhD ; Saeko Ishida

Elise Marsan

MD Catherine Miquel

MD Arnaud Biraben

; Dang

MD Khoa Nguyen

MD Doug Nordli

MD, PhD Patrick Cossette

MD Sylvie Nguyen

MD Virginie Lambrecq

MD Mihaela Vlaicu

Maïlys Daniau

MD, PhD Franck Bielle

MD, PhD Eva Andermann

MD Frederick Andermann

MD, PhD Eric Leguern

MD Francine Chassoux

MD Fabienne Picard
email: fabienne.picard@hcuge.ch

Familial focal epilepsy with focal cortical dysplasia due to DEPDC5 mutations

Objective:

The DEPDC5 (DEP domain-containing protein 5) gene, encoding a repressor of the mTORC1 signaling pathway, has recently emerged as a major gene mutated in familial focal epilepsies. We aimed to further extend the role of DEPDC5 to focal cortical dysplasias (FCDs).

Methods:

Seven patients from four families with DEPDC5 mutations, and focal epilepsy associated with FCD were recruited and investigated at the clinical, neuroimaging and histopathological levels. The DEPDC5 gene was sequenced from genomic blood and brain DNA.

Results: All patients had drug-resistant focal epilepsy, five of them underwent surgery, and one a brain biopsy. Electroclinical phenotypes were compatible with FCD II, although MRI was typical in only four cases. Histopathology confirmed FCD IIa in two patients (including one MRI-negative case) and showed FCD I in two other patients, and remained inconclusive in the last two. Three patients were seizure-free post-surgically and one had a worthwhile improvement. Sequencing of blood DNA revealed truncating DEPDC5 mutations in all four families; one mutation was found to be mosaic in an asymptomatic father. A brain somatic DEPDC5 mutation was identified in one patient in addition to the germline mutation.

Interpretation: Germline, germline mosaic and brain somatic DEPDC5 mutations may cause epilepsy associated with FCD, reinforcing the link between mTORC1 pathway and FCDs.

Similarly to other mTORopathies, a "two-hit" mutational model could be responsible for cortical lesions. Our study also indicates that epilepsy surgery is a valuable alternative in the treatment of drug-resistant DEPDC5-positive focal epilepsies, even if the MRI is unremarkable.

INTRODUCTION

Focal epilepsies are the most frequent epilepsies. It is increasingly recognized that focal epilepsies are likely to have a genetic basis. During the last two decades, a number of mutations have been identified in autosomal dominant focal epilepsies, in both ion-channel (CHRNA4, CHRNB2, CHRNA2, KCNT1) and non-ion channel genes (LGI1, DEPDC5). [START_REF] Baulac | Genetics advances in autosomal dominant focal epilepsies: focus on DEPDC5[END_REF][START_REF] Steinlein | Genetic heterogeneity in familial nocturnal frontal lobe epilepsy[END_REF] Focal cortical dysplasias (FCDs), in particular FCD II, are well-known underlying neuropathological conditions for severe therapy-refractory epilepsies. [START_REF] Muhlebner | Neuropathologic measurements in focal cortical dysplasias: validation of the ILAE 2011 classification system and diagnostic implications for MRI[END_REF][START_REF] Sisodiya | Malformations of cortical development: burdens and insights from important causes of human epilepsy[END_REF] Typically, FCDs II are characterized by a focal epilepsy of early onset, a drug-resistance, sleep-related seizures usually of extratemporal (often frontal) origin, and stereoelectroencephalography (SEEG) often showing subcontinuous repetitive spikes and spikes and waves alternating with bursts of fast activity. [START_REF] Chassoux | Stereoelectroencephalography in focal cortical dysplasia: a 3D approach to delineating the dysplastic cortex[END_REF][START_REF] Chassoux | Type II focal cortical dysplasia: electroclinical phenotype and surgical outcome related to imaging[END_REF][START_REF] Spreafico | Focal Cortical Dysplasias: clinical implication of neuropathological classification systems[END_REF] In the 2011 consensus classification of the International League Against Epilepsy (ILAE), FCD I referred to architectural disturbances of cortical lamination. Dysmorphic neurons are the histopathologic hallmark of FCD IIa and association with balloon cells specifies FCD IIb. [START_REF] Blumcke | The clinicopathologic spectrum of focal cortical dysplasias: a consensus classification proposed by an ad hoc task force of the ILAE diagnostic methods commission[END_REF] FCD II occurs mainly sporadically, but rare familial cases have been reported. [START_REF] Leventer | Is focal cortical dysplasia sporadic? Family evidence for genetic susceptibility[END_REF] Several rare mutations in CNTNAP2 (contactin associated protein-like 2), PTEN (phosphatase and tensin homolog) and SCN1A (alpha 1 subunit, sodium voltage-gated channel) have been associated with FCDs. [START_REF] Sisodiya | Focal cortical dysplasia type II: biological features and clinical perspectives[END_REF] Mutations of the DEPDC5 (DEP domain containing protein 5) gene account for 12 to 37 % of inherited focal epilepsies, including familial focal epilepsy with variable foci (FFEVF), autosomal dominant nocturnal frontal lobe epilepsy (ADNFLE) and familial temporal lobe epilepsy (FTLE). [START_REF] Dibbens | Mutations in DEPDC5 cause familial focal epilepsy with variable foci[END_REF][START_REF] Ishida | Mutations of DEPDC5 cause autosomal dominant focal epilepsies[END_REF][START_REF] Picard | DEPDC5 mutations in families presenting as autosomal dominant nocturnal frontal lobe epilepsy[END_REF] Recently, DEPDC5 mutations were also identified in rare cases of rolandic epilepsy and unclassified focal childhood epilepsies [START_REF] Lal | DEPDC5 mutations in genetic focal epilepsies of childhood[END_REF] , as well as in three patients (in two families) with bottom-of-the-sulcus dysplasia and one with focal band heterotopia. [START_REF] Scheffer | Mutations in mammalian target of rapamycin regulator DEPDC5 cause focal epilepsy with brain malformations[END_REF] Recent studies have shown that DEPDC5 is an upstream negative regulator of mammalian target of rapamycin mTOR complex 1 (mTORC1) in both mammalian cells [START_REF] Bar-Peled | A tumor suppressor complex with GAP activity for the Rag GTPases that signal amino acid sufficiency to mTORC1[END_REF] and yeast. [START_REF] Panchaud | Amino acid deprivation inhibits TORC1 through a GTPase-activating protein complex for the rag family GTPase Gtr1[END_REF] The mTORC1 signaling pathway regulates many major cellular processes including cell proliferation, growth and survival, and protein synthesis. [START_REF] Laplante | mTOR signaling in growth control and disease[END_REF] Upregulation of the pathway is implicated in the pathogenesis of numerous neurological conditions associated with cortical malformations and intractable seizures including FCDs IIb [START_REF] Lim | Focal malformations of cortical development: new vistas for molecular pathogenesis[END_REF] , TSC (tuberous sclerosis complex) [START_REF] Crino | Evolving neurobiology of tuberous sclerosis complex[END_REF] , PMSE (polyhydramnios, megalencephaly and symptomatic epilepsy) syndrome [START_REF] Parker | Rapamycin prevents seizures after depletion of STRADA in a rare neurodevelopmental disorder[END_REF] and HME (hemimegalencephaly). [START_REF] Lee | De novo somatic mutations in components of the PI3K-AKT3-mTOR pathway cause hemimegalencephaly[END_REF] Besides, a causal link between mTOR hyperactivation and epileptogenesis is well recognized. [START_REF] Lasarge | Mechanisms regulating neuronal excitability and seizure development following mTOR pathway hyperactivation[END_REF] Here, we confirm that mutations in mTORC1 repressor DEPDC5 cause drug-resistant focal epilepsy associated with FCDs. We describe the electroclinical and imaging phenotypes in seven patients from four families who were referred for epilepsy surgery, and report histological data and surgical outcomes.

SUBJECTS AND METHODS

Patients

Two French (families 1 and 2), one American (family 3) and one French-Canadian (family 4) families were recruited. Our criteria for inclusion in the cohort were: 1) the presence of at least two family members with focal epilepsy, 2) transmission compatible with autosomal dominant inheritance, 3) at least one family member with a diagnosis of FCD according to the electroclinical features with positive MRI, or negative MRI with histological confirmation, and 4) a pathogenic DEPDC5 mutation in the affected family members. Nocturnal frontal lobe epilepsy (NFLE) was defined by clusters of short-duration (less than one minute) hyperkinetic, tonic or dystonic seizures, occurring predominantly during sleep. Written informed consent was obtained from all participants (or the parents of minors).

Genetic study

Genomic DNA from blood samples was extracted using standard procedures. Genomic DNA from brain paraffin-embedded sections was available from 2 patients (1/IV-9 and 2/II-2), and was isolated and purified using the iPrep ChargeSwitch® Forensic Kit (Invitrogen™). All 42 coding exons and intron-exon junctions of DEPDC5 (except exon 2 which was analyzed by Sanger sequencing) were analyzed by universal tailed amplicon sequencing as previously described. [START_REF] Ishida | Mutations of DEPDC5 cause autosomal dominant focal epilepsies[END_REF] DEPDC5 cDNA refers to NCBI reference sequence # NM_001242896.1 and DEPDC5 protein to reference protein # NP_001229825.1. Ultra-deep sequencing (> 1000 X) to search for mosaic mutations in family 2 was performed on MiSeq (Illumina). Paternity test was performed using the AmpFLSTR® Profiler Plus® (Life technologies) in family 2.

RESULTS

Clinical and neuroimaging data (see Table 1, Figure 1) Family 1, diagnosed with FFEVF, was previously reported (referred to as family S) [START_REF] Ishida | Mutations of DEPDC5 cause autosomal dominant focal epilepsies[END_REF][START_REF] Picard | Dominant partial epilepsies. A clinical, electrophysiological and genetic study of 19 European families[END_REF] and was further investigated in this study. Patient 1/IV-9 had drug-resistant MRI-negative NFLE.

Intracranial recordings by SEEG demonstrated interictal, almost continuous, rhythmic spike activity co-localized with subclinical ictal discharges arising from the left lateral premotor cortex. Low frequency stimulation in the same area elicited three clinical seizures similar to the spontaneous ones. This electrophysiological pattern was highly suggestive of FCD II, as previously described. [START_REF] Chassoux | Stereoelectroencephalography in focal cortical dysplasia: a 3D approach to delineating the dysplastic cortex[END_REF] A focal resection guided by the results of the SEEG completely suppressed the seizures (13-year follow-up). However, due to technical problems, histopathology failed to identify the typical pattern corresponding to FCD II and only revealed A limited surgical resection was performed after SEEG at the age of 13, followed by a mild hemiparesis without reducing seizure frequency. Histopathology revealed fragmented and ischemic surgical specimens, precluding a reliable analysis (data not shown).

Germline DEPDC5 mutations

We applied targeted high-coverage sequencing of DEPDC5 exons and intron-exon junctions in the probands of the four families followed by a validation with Sanger sequencing. Pedigrees and molecular findings are illustrated in Figure 1. DEPDC5 c.715C>T (p.Arg239*) mutation was previously reported to segregate among affected family members in family 1, including patients 1/IV-9 and 1/V-1 who underwent epilepsy surgery. [START_REF] Ishida | Mutations of DEPDC5 cause autosomal dominant focal epilepsies[END_REF] We showed that the DEPDC5 transcript carrying the p.Arg239* mutation is targeted by nonsense-mediated decay and leads to DEPDC5 haploinsufficiency. [START_REF] Ishida | Mutations of DEPDC5 cause autosomal dominant focal epilepsies[END_REF] A novel mutation, c.484-1G>A, located in the acceptor splice site of intron 8 and predicted to induce exon 9 skipping, was identified in the proband (2/II-2) of family 2. The c.484-1G>A mutation was also present in the affected sister (2/II-1), but was undetectable by traditional Sanger sequencing in the parents, in whom paternity was confirmed. Deep-targeted sequencing of DEPDC5 exon 9 was subsequently performed in both parents, and allowed detection of the c.484-1G>A mutation in ~ 5% (56/1038 reads) of the alleles of the asymptomatic father (2/I-2), indicating germline mosaicism. In families 3 and 4, two novel nonsense mutations, c.1264C>T (p.Arg422*) and c.1759C>T (p.Arg587*) respectively, were identified. All the above mutations were considered pathogenic since they either introduced a premature stop codon or affected an essential splice site. They are not reported in the dbSNP135, 1000

Genomes Project database or the NHLBI exome variant server database.

Somatic DEPDC5 mutations

We searched for DEPDC5 somatic mutations in brain specimens from patients 1/IV-9 and 2/II-2. DNA was extracted from paraffin-embedded tissue and DEPDC5 sequencing performed either by targeted or traditional Sanger sequencing when amplification of targeted regions failed (exons 2, 12, 18). Sequencing confirmed the presence of the germline mutations p.Arg239* in patient 1/IV-9 and c.484-1G>A in patient 2/II-2. In addition, we also detected a second nonsense mutation located in exon 18, c.1264C>T (p.Arg422*) in individual 1/IV-9 (yet not detected in blood DNA with deep sequencing > 300 X) by Sanger sequencing (Fig 4). PCR amplification and sequencing were repeated from two independent extractions of brain tissue. The quality of brain DNA isolated from formalin-fixed, paraffinembedded tissue precluded deep sequencing to evaluate the level of mosaicism of the p.Arg422* mutation and to determine whether this second mutation was located in cis or trans from the p.Arg239* germline mutation. No brain somatic mutation was identified in the second patient (individual 2/II-2).

DISCUSSION

While DEPDC5 has been mainly linked to nonlesional familial focal epilepsies, particularly FFEVF, more recently mutations have also been detected in patients with focal epilepsy associated with cortical dysplasia and focal band heterotopia. [START_REF] Scheffer | Mutations in mammalian target of rapamycin regulator DEPDC5 cause focal epilepsy with brain malformations[END_REF] Thus, DEPDC5 mutations are associated with various focal epileptic phenotypes, across the spectrum from nonlesional familial focal epilepsies to malformation-associated focal epileptic syndromes (for review [START_REF] Baulac | Genetics advances in autosomal dominant focal epilepsies: focus on DEPDC5[END_REF][START_REF] Poduri | DEPDC5 does it all: shared genetics for diverse epilepsy syndromes[END_REF] ). Here, we confirm the existence of DEPDC5 mutations in patients with FCDs, and provide further information on the electroclinical, imaging and histological features, and on surgical We previously reported that DEPDC5 mutations may be associated with a high rate of drug-resistance. [START_REF] Picard | DEPDC5 mutations in families presenting as autosomal dominant nocturnal frontal lobe epilepsy[END_REF] We now report in seven patients from four families the existence of FCD in the context of inherited drug-resistant focal epilepsies caused by DEPDC5 mutations. All patients had electroclinical phenotypes highly suggestive of FCD II. Epilepsy onset in infancy and early childhood (mean age: 18 months), high seizure frequency, drug-resistance and predominant location in fronto-central areas are characteristic of FCD II. [START_REF] Chassoux | Type II focal cortical dysplasia: electroclinical phenotype and surgical outcome related to imaging[END_REF][START_REF] Mellerio | Optimizing MR imaging detection of type 2 focal cortical dysplasia: best criteria for clinical practice[END_REF] In addition, sleep-related epilepsy previously reported as a clinical marker of FCD II [START_REF] Nobili | Taylor's focal cortical dysplasia increases the risk of sleep-related epilepsy[END_REF] was observed in six out of seven patients. MRI showed typical features of FCD II in four patients and was considered normal in three others in accordance with the 30-40 % rate of MRI-negative FCD II cases observed in recent series. [START_REF] Mellerio | Optimizing MR imaging detection of type 2 focal cortical dysplasia: best criteria for clinical practice[END_REF][START_REF] Hong | Automated detection of cortical dysplasia type II in MRI-negative epilepsy[END_REF] Of the three MRI negative cases, one had macroscopic abnormalities suggestive of FCD II and two had SEEG features typical of FCD II, concordant with focal hypometabolism on FDG-PET in one. This is concordant with a report showing that in patients with MRI-negative neocortical epilepsies, FDG-PET coregistered on MRI may help to detect occult FCD II. [START_REF] Ryvlin | Epilepsy surgery in children and adults[END_REF][START_REF] Chassoux | FDG-PET improves surgical outcome in negative MRI Taylor-type focal cortical dysplasias[END_REF] In our series, histological examination confirmed FCD II in two patients, yet failed to demonstrate FCD II in four operated/biopsied patients, which may be due to technical reasons: fragmented specimens (individuals 1/IV-9, 1/V-1; 4/II-1), incomplete resection of the abnormal cortex (individuals 1/V-1, 4/II-1) and a biopsy (individual 3/IV-2), which was seemingly performed in the vicinity of the abnormal cortex.

Finally, of the five patients who underwent resective surgery, three became seizure-free with a follow-up of 4 to 13 years and one had a worthwhile improvement, in accordance with the favorable surgical outcome usually obtained in FCD II, including MRI-negative cases. [START_REF] Chassoux | Type II focal cortical dysplasia: electroclinical phenotype and surgical outcome related to imaging[END_REF] In the two patients in whom surgery did not lead to seizure freedom, the location of the epileptogenic zone in functional areas precluded complete removal of the lesion.

Despite a drug-resistance rate of ~ 30% in patients with familial focal epilepsies [START_REF] Picard | Dominant partial epilepsies. A clinical, electrophysiological and genetic study of 19 European families[END_REF][START_REF] Marini | The role of the nicotinic acetylcholine receptors in sleep-related epilepsy[END_REF] , which may be higher when DEPDC5 is causal [START_REF] Picard | DEPDC5 mutations in families presenting as autosomal dominant nocturnal frontal lobe epilepsy[END_REF] , epilepsy surgery is rarely performed. The reticence may derive from the absence of an MRI-detectable lesion and the presence of a mutated protein expressed ubiquitously throughout the cortex. Our study indicates that the presence of a DEPDC5 mutation in patients with refractory focal epilepsy (whether the brain MRI is normal or not) should not preclude epilepsy surgery, since it may cure the epilepsy.

Therefore we believe that drug-resistant focal epilepsies should be evaluated for surgery even in the presence of a clear genetic etiology. A key question directly emanating from our findings is whether only a subset of family members develops FCD or whether infraradiological FCD is present in all affected family members.

The finding that DEPDC5 is associated with FCD in the context of inherited focal epilepsy agrees with evidence that DEPDC5 acts as an mTORC1 inhibitor. [START_REF] Bar-Peled | A tumor suppressor complex with GAP activity for the Rag GTPases that signal amino acid sufficiency to mTORC1[END_REF] The role of DEPDC5 in the mTORC1 cascade is particularly attractive for two main reasons: 1/other genes encoding inhibitors of mTORC1 have been shown to cause brain malformations associated with refractory epilepsy [START_REF] Costa-Mattioli | mTOR complexes in neurodevelopmental and neuropsychiatric disorders[END_REF][START_REF] Crino | mTOR: A pathogenic signaling pathway in developmental brain malformations[END_REF] : inactivation of TSC1 or TSC2 causes tubers in TSC, deletion of STRADA causes PMSE syndrome, and loss-of-function of PTEN causes HME; 2/ enhanced constitutive mTORC1 activation has been shown in the balloon cells found in FCD IIb, which are thought to be the cellular marker traducing alterations of the pathway, but not in FCD IIa. [START_REF] Marin-Valencia | Pathogenetic mechanisms of focal cortical dysplasia[END_REF] Interestingly, we did not find FCD IIb cases in our families with DEPDC5 mutations, suggesting a continuum in both cellular and mTORC1 signaling alterations.

A long-standing, yet unproven hypothesis, is that FCD is caused by brain somatic mutations, which are increasingly identified in neurodevelopmental diseases associated with epilepsy. [START_REF] Poduri | Somatic mutation, genomic variation, and neurological disease[END_REF] Also, in families comprising a patient with Dravet syndrome, a severe epileptic encephalopathy, 7 % of SCN1A mutations are inherited from a parent who is somatic or germline mosaic. [START_REF] Depienne | Mechanisms for variable expressivity of inherited SCN1A mutations causing Dravet syndrome[END_REF] Interestingly, single cell sequencing of TSC1 and TSC2 in tubers revealed both germline and somatic mutations reflecting a "two-hit" mutational event. [START_REF] Crino | Biallelic TSC gene inactivation in tuberous sclerosis complex[END_REF] With the finding of DEPDC5 mutations in patients with cortical malformations, this hypothesis has become even more attractive. [START_REF] Poduri | DEPDC5 does it all: shared genetics for diverse epilepsy syndromes[END_REF] We report a de novo somatic DEPDC5 mutation in the brain lesional specimen of one patient with FCD. This mutation, absent from blood DNA, and superimposed on an existing, different, heterozygous germline mutation, is likely to cause a biallelic inactivation of DEPDC5 at the site of the lesion, confirming a two-hit mutational model. However, we failed to detect a somatic mutation in a second patient, from another family. Sequencing and search for loss-of-heterozygosity of paired blood-brain DNA from additional patients will allow determination of the prevalence of DEPDC5 somatic mutations in FCDs, and may help to resolve the puzzling question of the variability of brain areas involved in different family members within families with FFEVF. Sequencing using gene panels has been shown to be a valuable technique to detect mosaic mutations in cerebral cortical malformations. [START_REF] Jamuar | Somatic mutations in cerebral cortical malformations[END_REF] To conclude, our study reveals that DEPDC5 mutations are associated with FCD in patients with familial focal epilepsy, who were originally considered as nonlesional. DEPDC5, like other mTORC1 inhibitor genes including TSC1, TSC2, PTEN or STRADA, may be responsible for structural alterations in the cortex with the presence of dysmorphic cells.

Experimental studies on animal models are now needed to better understand the nature and extent of the consequent changes related to DEPDC5 mutations. In particular, it remains to be proven that constitutive mTORC1 activation contributes to DEPDC5-related epilepsy, as seen in other so-called mTORopathies. This would, of course, have important therapeutic implications.

  The study was approved by the local ethics committee (CCPPRB of Pitié-Salpêtrière Hospital, Paris, No. 69-03, 25/9/2003).HistopathologyHuman tissue specimens removed for the treatment of medically intractable epilepsy were analyzed by histology in five patients obtained from Rennes University (patients 1/IV-9 and 1/V-1), Saint-Anne Hospital in Paris (patient 2/II-2), Ann & Robert H. Lurie Children's Hospital of Chicago (patient 3/IV-2) and CHUM Notre-Dame in Montreal (patient 4/III-1). Brain specimens were fixed in 4% paraformaldehyde and embedded in paraffin. All fixed tissue blocks were sectioned at 8 μm. Histopathological examination of the cortical specimen was carried out on hematoxylin and eosin (HE) stained paraffin-embedded tissue. FCDs were classified according to the ILAE classification.[START_REF] Blumcke | The clinicopathologic spectrum of focal cortical dysplasias: a consensus classification proposed by an ad hoc task force of the ILAE diagnostic methods commission[END_REF]

a

  mild cytoarchitectonic perturbation compatible with FCD I (data not shown). His son, individual 1/V-4, suffered from nocturnal seizures suggestive of NFLE. A ketogenic diet in association with vigabatrin and phenytoin was effective for only 10 months. MRI showed abnormalities typical of FCD IIb in the left frontal lobe (Fig 2A-D). He is currently under presurgical evaluation. A second-degree female cousin of individual 1/V-4, individual 1/V-1, suffered from NFLE associated with an abnormal left precentral gyrus compatible with FCD II.

Family 2

 2 consisted of two sisters with nocturnal focal seizures. Patient 2/II-2 had drugresistant epilepsy and underwent a presurgical assessment. MRI was negative, but FDG-PET showed focal hypometabolism in the anterior part of the left insula (Fig 2E). SEEG confirmed the concordance between the hypometabolic area and the typical FCD II EEG pattern, with a seizure onset zone in the insula and early spread to the frontal lobe. After a limited resection of the left anterior insula, a seizure-free period of six months was obtained, but seizure recurrence required a complementary stereotaxic thermocoagulation within the neighboring anterior insular region, with a favorable outcome (4-year follow-up). Histopathological examination of a small fragment showed large dysmorphic neurons in favor of FCD IIa (Fig 3A-C). Her sister, individual 2/II-1, suffered from focal seizures suggestive of NFLE, sensitive to carbamazepine, which was discontinued two years later without seizure recurrence. Brain MRI (3 T) was normal. Family 3 included individual 3/IV-2 who had his first seizures at two years of age. Seizures were first sensitive to carbamazepine but a drug-resistance appeared at the age of five years. Semiology was suggestive of NFLE, MRI was normal, and presurgical evaluation implicated a precentral location. Because of the functional risks, surgery was limited to a surface biopsy from the suspect area. Histology revealed FCD I but the macroscopic anatomy (small area of pale cortex, firm to touch), suggested it may have been surrounded by FCD II (data not shown). Seizure frequency remained unchanged. Several family members on the father's side had a similar history of seizures (Fig 1), although all had spontaneous remission and did not participate in the study. Family 4 comprises a male and his nephew with high suspicion of FCD II in both. Individual 4/II-1 suffered from nocturnal sensorimotor seizures related to a dysplastic lesion located in the central region. Despite four resections, the patient was not seizure-free, but a worthwhile improvement was obtained. The excessive fragmentation of the surgical specimens precluded a reliable histologic diagnosis (data not shown). Individual 4/III-1 suffered from drug-resistant focal epilepsy from the age of four years (previously reported 25 ). Electroclinical features, including MRI, SPECT, FDG-PET and magnetoencephalography localized the epileptogenic zone to the right anterior frontal lobe (Fig 2F). A right anterior frontal corticectomy led to seizure freedom off all antiepileptic drugs (seven-year follow-up). Histopathological examination revealed FCD IIa (Fig 3B-D).

  outcome. Our findings have several original features: (1) the first report of patients with FCDs associated with DEPDC5 mutations who underwent epilepsy surgery, and demonstration of a favorable surgical outcome; (2) the presence of histologically confirmed FCD despite a normal MRI, raising the question of potential underestimation of FCDs in genetic focal epilepsies; (3) the association between FCD type IIa and DEPDC5 mutations suggesting that mTORC1 hyperactivation predicted in patients with mutations in DEPDC5 may also underlie different FCD subtypes, and not only type IIb; (4) the first report of a brain somatic mutation in DEPDC5 gene in one patient (out of two tested) in addition to an existing germline mutation supporting a two-hit hypothesis for FCD, and (5) a DEPDC5 germline mosaic mutation transmitted by an asymptomatic father.

Figure 1 :

 1 Figure 1: Pedigrees of the families with DEPDC5 mutation segregation. Pedigrees of family 1 (p.Arg239*) and family 2 (c.484-1G >A) of French origin, family 3 (p.Arg422*) of American origin and family 4 (p.Arg587*) of French-Canadian origin are illustrated. Individuals with FCD are boxed (in red). The respective DEPDC5 mutations (NM_001242896.1, NP_001229825.1) are indicated. Individuals who carry a DEPDC5 mutation are denoted by m/+ and those negative for mutations are denoted by +/+. Level of mosaicism of c.484-1G >A mutation is indicated in brackets.

Figure 2 :

 2 Figure 2: MRI of patients 1/V-4 (A-D), 2/II-2(E) and 4/III-1 (F). (A, B), coronal slices, inverted T2 weighted sequences; (C, D) axial slices; C, T1 weighted; and D, FLAIR sequences. Typical features of FCD IIb located in the left mesial superior frontal gyrus (premotor area) with a broad gyrus, blurring of the grey-white matter demarcation, subcortical hypersignal and transmantle sign (arrows). (E), FDG-PET/MRI superimposition, coronal slice; note the focal hypometabolism in the anterior part of the left insula (arrow) corresponding to the seizure onset zone and the site of FCD IIa. (F), MRI of individual 4/III-1 showing a coronal view, 3D T1 sequence. The white arrow shows the abnormal deep sulcus (FCD IIa) in the paramedian region of the right frontal pole.

Figure 3 :

 3 Figure 3: Hematoxylin and eosin stained paraffin-embedded tissue from post-operative brain sections in subject 2/II-2 (A-C) and 4/III-1 (B-D), showing the presence of large dysmorphic neurons (indicated by arrows) and cortical disorganization characteristic of FCD IIa. Scale bar = 50 µm, magnification A-B: X40, C-D: X20.

Figure 4 :

 4 Figure 4: Sequence chromatograms from patient 1/IV-9 showing the detection of p.Arg239* mutation in DEPDC5 exon 12 in blood and brain DNA, and p.Arg422* mutation in exon 18 in brain DNA. The mutation p.Arg422* was not detected from blood DNA indicating that it occurred de novo somatically.

  Figure 1

Figure 4

 4 Figure 4

ACKNOWLEDGMENTS

We thank the patients and families for their participation in the study. We thank the genotyping and sequencing ICM platform (Y. Marie), bioinformatics/biostatistics core facility of ICM (J. Guegan), the DNA and cell bank ICM platform (C. Dussert), M. Giry for technical assistance, and Richard Miles and Sarah Weckhuysen for helpful discussion. The research is supported by the funds from the Swiss National Foundation No. 320030-127608 (FP), Fondation pour la recherche sur le cerveau (SB), Fondation Française pour la recherche sur les épilepsies (SB), Carnot Institute and the program "Investissements d'avenir" ANR-10-IAIHU-06. The authors have no potential conflicts of interest.