Colette De Coster
email: colette.decoster@univ-valenciennes.fr

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

Singular behavior of the solution of the Cauchy-Dirichlet heat equation in weighted L p -Sobolev spaces

Keywords: heat equation, singular behavior, nonsmooth domains. AMS Subject Classification: 35K15, 35B65

We consider the heat equation on a polygonal domain Ω of the plane in weighted

Here h belongs to L p (0, T ; L p µ (Ω)), where L p µ (Ω) = {v ∈ L p loc (Ω) : r µ v ∈ L p (Ω)}, with a real parameter µ and r(x) the distance from x to the set of corners of Ω. We give sufficient conditions on µ, p and Ω that guarantee that problem (0.1) has a unique solution u ∈ L p (0, T ; L p µ (Ω)) that admits a decomposition into a regular part in weighted L p -Sobolev spaces and an explicit singular part.

Introduction

In this work we consider the Cauchy-Dirichlet problem for the heat equation (0.1) on a polygonal domain Ω of the plane. We give the singular decomposition of the solution of (0.1) in weighted L p -Sobolev spaces with precise regularity information on the regular and singular parts. The classical Fourier transform techniques do not allow to handle such a general case. Hence we use the theory of sums of operators as in G. Da Prato and P. Grisvard [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] and G. Dore and A. Venni [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF]. These results have been fruitfully used to prove the singular behavior of elliptic problems in non-Hilbertian Sobolev spaces in [START_REF] Grisvard | Singular behavior of elliptic problems in non hilbertian Sobolev spaces[END_REF].

Although the analysis of the heat equation is well developed in weighted L 2 -Sobolev spaces [START_REF] Grisvard | Edge behavior of the solution of an elliptic problem[END_REF][START_REF] Kozlov | Singularities of solutions of the first boundary value problem for the heat equation in domains with conical points[END_REF][START_REF] Kozlov | Coefficients in the asymptotic solutions of the Cauchy boundary-value parabolic problems in domains with a conical point[END_REF][START_REF] Anh | Asymptotic formulas for solutions of parameterdepending elliptic boundary-value problems in domains with conical points[END_REF] or in L p -Sobolev spaces [START_REF] Grisvard | Singular behavior of elliptic problems in non hilbertian Sobolev spaces[END_REF], to the best of our knowledge such a singularity result does not exist in the framework of weighted L p -Sobolev spaces. For maximal regularity type results in weighted L p -Sobolev spaces, we refer to [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF][START_REF] Nazarov | L p -estimates for the solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension[END_REF][START_REF] Solonnikov | L p -estimates for solutions of the heat equation in a dihedral angle[END_REF][START_REF] Nazarov | Dirichlet problem for quasilinear parabolic equations in domains with smooth closed edges[END_REF][START_REF] Prüss | H ∞ -Calculus for the sum of non-commuting operators[END_REF].

In [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF], we have considered the same kind of results for the periodic-Dirichlet problem

∂ t u -∆u = g, in Ω × ] -π, π[, u = 0, on ∂Ω × [-π, π], u(•, -π) = u(•, π), in Ω.
Some of the results presented there are useful in our context too.

The first step, which consists in the study of the Helmholtz equation

-∆u + zu = g, in Ω, u = 0, on ∂Ω, (1.1)
where z is a complex number, was performed in [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF].

The paper is organized as follows: In section 2 we apply the approach of Da Prato-Grisvard [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] to obtain a decomposition but with non-optimal regularity informations. Section 3 is devoted to the proof of the regularity of (∂ t -∆)S, where S is the singular part of the solution obtained before. The use of the approach of Dore-Venni [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF] and the results from section 3 allows to get the optimal regularity result in section 4.

In the whole paper the notation a b means the existence of a positive constant C, which is independent of the quantities a, b (and eventually of the above parameter z) under consideration such that a ≤ Cb.

2 Application of Da Prato-Grisvard's approach [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] Let us assume in the future that the assumptions of [6, Theorem 2.3] are satisfied, i.e., (H) Let p ≥ 2 and Ω be a bounded polygonal domain of R 2 , i.e., its boundary is the union of a finite number of line segments. Denote by S j , j = 1, . . . , J, the vertices of ∂Ω enumerated clockwise and, for j ∈ {1, 2, . . . , J}, let ψ j be the interior angle of Ω at the vertex S j and λ j = π ψ j . For all j = 1, . . . , J, let µ j > -λ j satisfy 2 -2 p -µ j = kλ j , for all k ∈ Z * , and

µ j < 2p -2 p , if p > 2, µ j ≤ 1, if p = 2; |µ j | < 2 √ p -1 p λ j . (2.1)
We shall apply the results from [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] (see also [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]Theorem 2.1]) on the space

E = L p (I; L p µ (Ω)) with L p µ (Ω) = {f ∈ L p loc (Ω) | wf ∈ L p (Ω)}, where I = [0, T ], w(x)
r(x) µ j near S j , w(x) 1 far from the corners and with the operators

A : D(A) ⊂ E → E : u → -∆u, with D(A) = L p (I; D(∆ p, µ )) where D(∆ p, µ ) = {u ∈ H 1 0 (Ω) | ∆u ∈ L p µ (Ω)}, and
B T : D(B T ) ⊂ E → E : u → ∂ t u, with D(B T ) = W 1,p left (I; L p µ (Ω)) = {u ∈ E | ∂ t u ∈ E, u(•, 0) = 0}. Proposition 2.1.
Under assumptions (H), the operator A + B T has an inverse closure i.e., for all g ∈ L p (I; L p µ (Ω)), there exists a unique strong solution u ∈ L p (I; L p µ (Ω)) of (A + B T )u = g i.e. there exists

(u n ) n ⊂ D(A) ∩ D(B T ) such that u n → u and Au n + B T u n → g. Moreover we have u = 1 2πi γ (A + z I) -1 (z I -B T ) -1 g dz, (2.2)
with γ : R → C defined for example by γ(s)

= |s| e -i( π 2 +δ) for s ≤ 0, γ(s) = |s| e i( π 2 +δ) for s > 0, with δ ∈ ]0, θ A -π 2 [ and θ A ∈] π 2 , π[ given by [6, Theorem 2.3].
Proof. The proof follows the lines of [6, Proposition 3.1] with minor changes concerning B T : a simple calculation proves that ρ(B T ) = C and, in the verification that, for all θ B < π 2 , there exists M ≥ 0 such that, for all µ

∈ S B T = {µ ∈ C | | arg(µ)| ≤ θ B }, (B T + µ I) -1 ≤ M |µ| -1 , denoting v = w p |u| p-2 ū, we have to replace p 2 Ω π -π v∂ t u dtdx + Ω π -π v∂ t u dtdx = 0, valid in the periodic case, by p 2 Ω T 0 v∂ t u dtdx + Ω T 0 v∂ t u dtdx = Ω |u(x, T )| p w(x) p dx.
The remainder of the proof follows in the same way as in [6, Proposition 3.1].

Remark 2.1 As in [6, Remark 3.1], we obtain also

(1 + |z|) (z I -B T ) -1 g L p (I;L p µ (Ω)) g L p (I;L p µ (Ω)) .
As it is clear that, for each t, we have

[(A + z I) -1 h](t) = (-∆ + z I) -1 (h(t)),
we can use the decomposition in regular and singular parts of the solution of the Helmholtz equation (1.1) obtained in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF] (see [6, (2.4)]) and rewrite (2.2) as

u = u R + J j=1 η j k∈N:0<λ j =kλ j <2-2 p -µ j u λ j , (2.3)
where u R = 1 2πi γ R(z)(z I -B T ) -1 g dz, u λ j = 1 2πi γ T λ j (z), (z I -B T ) -1 g ψλ j ,z dz, (2.4) with R(z) : L p µ (Ω) → V 2,p µ (Ω)
the operator which gives the regular part of the solution of (1.1), T λ j (z) :

L p µ (Ω) → C : g → c λ j (z) = T λ j (z)
, g the one which gives the singular coefficient of the solution of (1.1); η j is a radial cut-off function such that η j ≡ 1 in a small ball centered at S j and η j ≡ 0 outside a larger ball; P j,λ j (s) =

l j,λ j -1 i=0 s i i! with l j,λ j > 2 -µ j -2 p -λ j and ψλ j ,z (r, θ) = P j,λ j (r √ z)e -r √ z r λ j sin(λ j θ). Recall that V 2,p µ (Ω) is defined as the closure of C ∞ S (Ω) = {v ∈ C ∞ (Ω) | S j ∈ supp v} with respect to the norm u V 2,p µ (Ω) = ( |γ|≤2 Ω |D γ u(x)| p w p (x) r (|γ|-k)p (x) dx) 1/p .
For more details, see [6, end of Section 2].

Proposition 2.2. Let the assumptions (H) be satisfied and denote σ j := 1 -1 p -µ j +λ j 2 . Then for all s ∈ ]0, min(1 -σ j , 1/p)[, for all g ∈ W s,p (I, L p µ (Ω)), there exist q λ j ∈ W s+σ j ,p (I) and E λ j such that u λ j defined by (2.4) can be written as u λ j = (E λ j * t q λ j ) r λ j sin(λ j θ).

(2.5) Moreover we have

q λ j = 1 2πi γ T λ j (z), (z I -B T ) -1 g dz, E λ j (x, t) = 1 2π R e iξt P j,λ j (r √ iξ) e -r √ iξ dξ,
(2.6) and the operator U : W s,p (I, L p µ (Ω)) → W s+σ j ,p (I) : g → q λ j is continuous.

Proof. Recall that for all f ∈ L p µ (Ω), the mapping

C → C : z → T λ j (z), f is holomor- phic on A := {z ∈ C | | arg(z)| < θ A }
and continuous on A (see [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]).

Step 1: Extension. Let us consider the extension of g to Ω × R, defined by

g(x, t) = g(x, t) if t ∈ [0, T ], g(x, t) = 0 if t ∈ [0, T ],
and denote by ũz = (z I -B ∞ ) -1 g, the solution of

z ũ -∂ t ũ = g, in Ω × R, ũ(•, 0) = 0, in Ω.
Observe that, by uniqueness of the solution of the Cauchy problem, we have ũz

| [0,T ]×Ω = (z I -B T ) -1 g. Moreover we easily see that ũz (x, t) = 0, if t < 0, = - t 0 e z(t-s) g(x, s) ds, if t ∈ [0, T ],
= -e zt T 0 e -zs g(x, s) ds, if t > T.

Consider the function ũλ

j (x, t) = 1 2πi γ T λ j (z), (z I -B ∞ ) -1 g ψλ j ,z (r, θ) dz. (2.7)
Observe that ũλ j | Ω×[0,T ] = u λ j and that, for t > T , z = ρe ±iθ 0 with ρ > 0, θ 0 = π 2 + δ, using [6, (2.6)], we have

T λ j (z), (z I -B ∞ ) -1 g ψλ j ,z (r, θ) T λ j (z), (z I -B ∞ ) -1 g |e z(t-T ) | T λ j (z) (L p µ (Ω))
T 0 e z(T -s) g(x, s) ds

L p µ (Ω) e -ρ| cos θ 0 |(t-T ) 1 1 + ρ σ j T 0 e -qρ| cos θ 0 |(T -s) ds 1/q g L p (0,T ;L p µ (Ω))
e -ρ| cos θ 0 |(t-T ) 1 1 + ρ σ j g L p (0,T ;L p µ (Ω))

On the other hand, for 0 < t < 2T and |z| = ρ we have, by Remark 2.1,

| T λ j (z), (z I -B ∞ ) -1 g ψλ j ,z (r, θ)| | T λ j (z), (z I -B ∞ ) -1 g | | T λ j (z), (z I -B 2T ) -1 g | 1 1 + ρ σ j 1 1 + ρ g L p (0,T ;L p µ (Ω)) .
Step 2: For all x ∈ Ω, the function ũλ j (x, •) ∈ L 2 (R) and hence admits a partial Fourier transform in t. For all t > 2T by the previous considerations, we have

|ũ λ j (x, t)| γ T λ j (z), (z I -B ∞ ) -1 g ψλ j ,z (r, θ) dz ∞ 0 e -ρ| cos θ 0 |(t-T ) dρ g L p (0,T ;L p µ (Ω)) 1 t -T g L p (0,T ;L p µ (Ω)
) . For t < 2T we use a similar argument using here the last estimate of Step 1. This shows that, for all x ∈ Ω, ũλ j (x, •) ∈ L 2 (R), and we can take its partial Fourier transform in t.

Step 3: The partial Fourier transform in t of ũλ j (x, •) satisfies, for all ξ = 0,

F t (ũ λ j )(x, ξ) = -T λ j (iξ), F t (g)(•, ξ) ψλ j ,iξ (x).
As ũλ j (x, •) ∈ L 2 (R), using [17, Cor 1, p.154], we know that

F t (ũ λ j )(x, ξ) = lim k→∞ k -k e -itξ ũλ j (x, t) dt.
Hence by the above computations we have, for k > 2T , k -k R T λ j (ρe i sgn(ρ)θ 0 ), (ρe i sgn(ρ)θ 0 I -B ∞ ) -1 g ψλ j ,ρe i sgn(ρ)θ 0 (x)e -iξt e i sgn(ρ)θ 0 dρ dt

2T 0 +∞ 0 1 1 + ρ σ j 1 1 + ρ dρ dt + k 2T +∞ 0 1 1 + ρ σ j e -ρ | cos θ 0 |(t-T ) dρ dt g L p (0,T ;L p µ (Ω)) 2T 0 +∞ 0 1 1 + ρ σ j 1 1 + ρ dρ dt + k 2T 1 | cos θ 0 |(t -T ) dt g L p (0,T ;L p µ (Ω)) < +∞.
Hence, by Fubini's theorem, we obtain

F t (ũ λ j )(x, ξ) = 1 2πi γ T λ j (z), F t ((zI -B ∞ ) -1 g)(•, ξ) ψλ j ,z (x) dz = 1 2πi γ T λ j (z), F t (g)(•, ξ) z -iξ ψλ j ,z (x) dz.
The rest of the proof follows [6, Step 2 of the Proof of Proposition 3.2] observing that, by Hölder inequality, we have

F t (g)(•, ξ) p L p µ (Ω) = Ω w p (x) R e -iξt g(x, t) dt p dx Ω w p (x) R |g(x, t)| dt p dx Ω w p (x) T 0 |g(x, t)| dt p dx g p L p (I;L p µ (Ω)) .
Step 4: The operator U : W s,p (I; L p µ (Ω)) → W s+σ j ,p (I) : g → q λ j with q λ j given by (2.6) is continuous. By the results of [START_REF] Grisvard | Equations différentielles abstraites[END_REF], as 0 < s < 1/p, we know that

W s,p (I; L p µ (Ω)) = g ∈ E | ∞ 0 ρ sp B T (B T -ρe ±i( π 2 +δ) I) -1 g p E ρ -1 dρ < ∞ .
We have a similar characterization of W s+σ j ,p (I) by considering the operator

N : D(N ) ⊂ L p (I) → L p (I) : u → ∂ t u with D(N ) = {u ∈ W 1,p (I) | u(0) = 0}.
Hence if s + σ j < 1/p, we have

W s+σ j ,p (I) = g ∈ L p (I) | ∞ 0 τ (s+σ)p N (N + τ I) -1 g p L p (I) τ -1 dτ < ∞ , while if s + σ j > 1/p, defining W s+σ j ,p left (I) = {g ∈ W s+σ j ,p (I) | g(0) = 0}, we have W s+σ j ,p left (I) = g ∈ L p (I) | ∞ 0 τ (s+σ)p N (N + τ I) -1 g p L p (I) τ -1 dτ < ∞ .
Claim 1: For τ ≥ 0, we have

N (N + τ I) -1 q λ j = 1 2πi γ T λ j (z), B T (zI -B T ) -1 g dz z + τ . (2.8)
First observe that

N (N + τ I) -1 q λ j = 1 2πi γ T λ j (z), B T (B T + τ I) -1 (zI -B T ) -1 g dz = 1 2πi γ T λ j (z), B ∞ (B ∞ + τ I) -1 (zI -B ∞ ) -1 g dz Ω×[0,T ]
.

Let us show that we can take the Fourier transform in

t of 1 2πi γ T λ j (z), B ∞ (B ∞ + τ I) -1 (zI -B ∞ ) -1 g dz.
We have

B ∞ (B ∞ + τ I) -1 (zI -B ∞ ) -1 g = (zI -B ∞ ) -1 g -τ (B ∞ + τ I) -1 (zI -B ∞ ) -1 g =: ṽz (x, t) -τ ṽzτ (x, t).
Observe that, for t > T , we have ṽz (x, t) = -e z(t-T )

T 0 e z(T -s) g(x, s) ds and ṽzτ (x, t) = -

T 0 e -τ (t-s) s 0 e z(s-σ) g(x, σ) dσds - t T e -τ (t-s) e z(s-T ) T 0 e z(T -σ) g(x, σ) dσds = - T 0 e z(t-σ) -e -τ (t-σ) z + τ g(x, σ) dσ.
Hence, for τ ≥ 0 and if t > 2T we have as above, using [6, (2.6)],

1 2πi γ T λ j (z), B ∞ (B ∞ + τ I) -1 (zI -B ∞ ) -1 g dz 1 t -T g L p (I;L p µ (Ω)) + γ τ |e z(t-T ) | |z + τ | T λ j (z), T 0 e z(T -σ) g(x, σ) dσ dz + γ τ |e -τ (t-T ) | |z + τ | T λ j (z), T 0 e -τ (T -σ) g(x, σ) dσ dz ≤ 1 t -T + γ τ |z + τ | |e z(t-T ) | dz + τ e -τ (t-T ) γ 1 (|z + τ |)(1 + |z| σ j ) dz g L p (I;L p µ (Ω)) 1 t -T + 1 sin θ 0 1 | cos θ 0 | 1 t -T +τ e -τ (t-T ) ∞ 1 1 1 + ρ σ j 1 ρ sin θ 0 dρ + e -τ (t-T ) sin θ 0 g L p (I;L p µ (Ω)) .
We conclude that this function belongs to L 2 (R, L p µ (Ω)) and we can take its Fourier transform in t. By Cauchy theorem, we obtain, as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF], that its Fourier transform in t is given by

-T λ j (iξ), F t (g)(•, ξ) iξ iξ + τ . (2.9)
In the same way, we can take the Fourier transform in t of the right-hand side of (2.8) since, for t > 2T we have

1 2πi γ T λ j (z), B ∞ (zI -B ∞ ) -1 g(x, t) dz z + τ = - 1 2πi γ ze z(t-T ) T λ j (z), T 0 e z(T -s) g(x, s) ds dz z + τ 1 t -T .
Hence by Cauchy theorem, as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF], its Fourier transform in t is given by (2.9). As the Fourier transform of the two functions coincide, the two functions are equal. Claim 2: For 0 < s < min(1 -σ j , 1/p), the operator U : W s,p (I; L p µ (Ω)) → W s+σ j ,p (I) : g → q λ j is continuous. The proof is the same as the corresponding one in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF].

Conclusion. By

Step 3, we have, for all ξ = 0,

F t (ũ λ j )(x, ξ) = -T λ j (iξ), F t (g)(•, ξ) ψλ j ,iξ (x).
(2.10)

Let qλ j (t) = 1 2πi γ T λ j (z), (z I -B ∞ ) -1 g(•, t) dz. (2.11)
As previously we can take its Fourier transform and we see, applying again the Cauchy theorem as above, that its Fourier transform is given by

F(q λ j )(ξ) = 1 2πi γ T λ j (z), F t (g)(•, ξ) z -iξ dz = -T λ j (iξ), F t (g)(•, ξ) .
Consider the function E λ j (x, t) which has as Fourier transform in t

F t (E λ j )(x, ξ) = P j,λ j (r √ iξ) e -r √ iξ .
As P j,λ j (r √ iξ) e -r √ iξ ∈ L ∞ (R) and by [18, p.113], L ∞ (R) ⊂ S , we have also by [START_REF] Zuily | Eléments de distributions et d'équations aux dérivées partielles[END_REF]p.114] that E λ j (x, •) ∈ S . Now observe that by [18, p.112], S ⊂ D . As qλ j ∈ L 2 (R), there exists a sequence (q n ) n ⊂ D(R) such that q n → qλ j in L 2 (R). By [START_REF] Zuily | Eléments de distributions et d'équations aux dérivées partielles[END_REF]Thm 6.3,p.120] or [START_REF] Yosida | Functional Analysis, Fourth Edition[END_REF]Thm 6,p.160], as F t (E λ j ) is bounded, we have that

F t (E λ j * q n ) = F t (E λ j ) F t (q n ) → F t (E λ j ) F t (q λ j ), in L 2 (R).
Hence, we have E λ j * q n → E λ j * qλ j , in L 2 (R), which proves that ũλ j = (E λ j * t qλ j ) r λ j sin(λ j θ) and the result follows.

As in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF] we can extend the previous Proposition to g ∈ L p (I, L p µ (Ω)).

Theorem 2.3. Let the assumptions (H) be satisfied and denote σ j := 1 -1 p -

µ j +λ j 2 .
Then for all g ∈ L p (I, L p µ (Ω)), the problem (0.1) has a unique strong solution u which can be written in the form

u = u R + J j=1 η j k∈N:0<λ j =kλ j <2-2 p -µ j u λ j ,
where u R (resp. u λ j ) is given by (2.4) (resp. (2.5)) with q λ j ∈ W σ j ,p (I) and E λ j given by (2.6). Moreover the mapping L p (I, L p µ (Ω)) → W σ j ,p (I) : g → q λ j is continuous.

3 Regularity of q λ j → ( ∂ ∂t -∆)(η j u λ j )

In order to consider the regularity of u R we observe that u R satisfies

∂ t u R -∆u R = g - J j=1 k∈N:0<λ j =kλ j <2-2 p -µ j (∂ t (η j u λ j ) -∆(η j u λ j )). (3.1)
Hence we need informations on the regularity of ∂ t (η j u λ j ) -∆(η j u λ j ). This is the aim of this section.

where M (t) = 1 if t > 0 and M (t) = 0 if t < 0. Recall that E is a tempered distribution. We easily check that the partial Fourier transform F t E in t of E is given by

F t E(r, ξ) = e -|r| √ iξ 2 √ iξ . As F t ( ∂ 2 ∂r 2 E) = ∂ 2 ∂r 2 (F t E) = √ iξ 2 e -|r| √ iξ -δ 0 (r) = F t ( H(|r|, t) 2 -δ 0 (r)δ 0 (t)),
and since F t is an isomorphism from S (R 2 ) into itself, we deduce that

H(|r|, t) = 2 ∂ 2 ∂r 2 E(r, t) + 2δ 0 (r)δ 0 (t).
Hence, for r > 0, H(r, t) = 2 ∂ 2 E ∂r 2 (r, t) and we conclude as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]. Theorem 3.2. Under assumptions (H) and recalling that σ j = 1 -1 p -µ j +λ j 2 , the mapping q λ j → ( ∂ ∂t -∆)(η j u λ j ) is continuous from W σ j ,p (I) into L p (I; L p µ (Ω)).

Proof. Recall that, by [6, Remark 3.2], 0 < σ j < 1.

Case 1: P j,λ j ≡ 1 i.e. λ j + µ j -1 + 2 p > 0. As in the proof of Proposition 2.2, consider the functions qλ j given by (2.11) and ũλ j given by (2.7).

Let us take the Fourier transform in t of f (x, t) = η j (r)( ∂ ∂t -∆)ũ λ j (x, t). We obtain F t f (x, ξ) = η j (r) F t (( ∂ ∂t -∆)ũ λ j ) = η j (r) (iξ I -∆) F t (ũ λ j ).

As in Step 3 of the proof of Proposition 2.2, we have F t (ũ λ j )(x, ξ) = -T λ j (iξ), F t (g)(•, ξ) ψλ j ,iξ (x). and hence (iξ I -∆) F t (ũ λ j ) = -c λ j (iξ) (iξ I -∆) (e -r √ iξ r λ j sin(λ j θ))

= -c λ j (iξ) √ iξ e -r √ iξ r λ j -1 sin(λ j θ) (2λ j + 1), with c λ j (iξ) = T λ j (iξ), F t (g)(•, ξ) = -F t (q λ j )(ξ). Using the kernel H given by (3.2), as previously, we obtain that f (x, t) = (H * t qλ j )(r) (2λ j + 1) r λ j -1 sin(λ j θ) η j (r). As = 0, we have f (x, t) = (2λ j + 1) r λ j -1 sin(λ j θ) η j (r) R H(r, s) [q λ j (t -s) -qλ j (t)] ds.

¿From this point on, the proof proceeds as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]. Case 2: deg(P j,λ j ) = l j,λ j -1 ≥ 1. This case is treated as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF] using Lemma 3.1.

4 Application of Dore-Venni's approach [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF] Now we are able to consider the regularity of u R and to prove our main result.

R

  H(r, s) ds= R e -itξ H(r, t) dt ξ=0 = F t H(r, 0) = iξe -r √ iξ ξ=0

  Proof. Let E be the elementary solution of the heat equation in R 2 , i.e.,

			E(r, t) =	M (t) √ 4πt	e -r 2 4t ,
		H(r, t) =	1 2π R	iξ e -r	√	iξ e iξt dξ	(3.2)
	satisfies, for all ∈ N,	∂ ∂r	H(r, t)	(|t| + r 2 ) -3+ 2 .	(3.3)

Lemma 3.1. The kernel H defined on R + × R by

Theorem 4.1. Let p ≥ 2, Ω be a bounded polygonal domain of R 2 . Denote by S j , j = 1, . . . , J, the vertices of ∂Ω enumerated clockwise and, for j ∈ {1, 2, . . . , J}, let ψ j be the interior angle of Ω at the vertex S j and λ j = π ψ j . For all j = 1, . . . , J, let µ j satisfies

and, for all k ∈ Z * , 2 -2 p -µ j = kλ j and µ j + kλ j = 1. Let σ j = -µ j +λ j 2

+ 1 -1 p , then, for every g ∈ L p (0, T ; L p µ (Ω)), there exists a unique solution u ∈ L p (0, T ;

Moreover u admits the decomposition

where q λ j ∈ W σ j ,p (I) and

Proof. As in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF], we prove that u R defined by (2.4) satisfies, for all θ ∈ ]0, 1[,

. We now observe that u R is a strong solution of (3.1) with a right-hand side in L p (I; L p µ (Ω)) according to the previous results. Then we apply Dore-Venni's approach [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF] (see also Theorem 2.2 of [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]) with E = L p (I; L p µ (Ω)), and

). The assumptions (H 3 ), (H 4 ), (H 5 ) of [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF] can be verified as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]. To verify (H 6 ) we apply the following result of Coifman -Weiss (see [START_REF] Coifman | Transference methods in analysis[END_REF] or for example [START_REF] Amann | Linear and quasilinear parabolic problems[END_REF]). If -C is the infinitesimal generator of a strongly continuous contraction semi-group in E which preserves the positivity then there exists K > 0 such that, for all s ∈ R,

For what concerns the operator A, the argument is the same as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF]. For what concerns B, we already know that -B is the generator of a C 0 semigroup of contractions S. It remains to verify that S preserves the positivity. As usual it suffices to check that its resolvent preserves positivity: Namely for λ > 0 consider the solution u ∈ D(B) of

Then u(x, t) = (B + λ I) -1 f = t 0 e -λ(t-s) f (x, s) ds which is clearly non negative.

We conclude as in [START_REF] Coster | Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces[END_REF] that u R ∈ L p (I; V 2,p µ (Ω)) ∩ W 1,p (I; L p µ (Ω)).


Colette De Coster

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

Colette Decoster

Singular behavior of the solution of the periodic-Dirichlet heat equation in weighted L p -Sobolev spaces

Keywords: heat equation, singular behavior, nonsmooth domains. AMS Subject Classification: 35K15, 35B65

We consider the heat equation in a polygonal domain Ω of the plane in weighted L p -Sobolev spaces

with a real parameter µ and r(x) the distance from x to the set of corners of Ω. We give sufficient conditions on µ, p and Ω that guarantee that problem (0.1) has a unique solution u ∈ L p (-π, π; L p µ (Ω)) that admits a decomposition into a regular part in weighted L p -Sobolev spaces and an explicit singular part.

The classical Fourier transform techniques do not allow to handle such a general case. Hence we use the theory of sums of operators.

Introduction

This paper is the second one of a large program of research devoted to the study of (nonlinear) heat equation in nonsmooth domains in weighted L p -Sobolev spaces. Our final goal requires precise information about the solution of the linear heat equation (1.1) in weighted L p -Sobolev spaces. In particular its decomposition into a regular part and an explicit singular part is needed. Although this theory is well developed in weighted L 2 -Sobolev spaces [START_REF] Grisvard | Edge behavior of the solution of an elliptic problem[END_REF][START_REF] Kozlov | Singularities of solutions of the first boundary value problem for the heat equation in domains with conical points[END_REF][START_REF] Kozlov | Coefficients in the asymptotic solutions of the Cauchy boundary-value parabolic problems in domains with a conical point[END_REF][START_REF] Anh | Asymptotic formulas for solutions of parameter-depending elliptic boundary-value problems in domains with conical points[END_REF] or in L p -Sobolev spaces [START_REF] Grisvard | Singular behavior of elliptic problems in non hilbertian Sobolev spaces[END_REF], to our best knowledge such a result does not exist in the framework of weighted L p -Sobolev spaces. The first class of papers are based on the Fourier analysis, while the second one uses the theory of sums of operators. For maximal regularity type results in weighted L p -Sobolev spaces, we refer to [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF][START_REF] Nazarov | L p -estimates for the solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension[END_REF][START_REF] Solonnikov | L p -estimates for solutions of the heat equation in a dihedral angle[END_REF][START_REF] Nazarov | Dirichlet problem for quasilinear parabolic equations in domains with smooth closed edges[END_REF][START_REF] Prüss | H ∞ -Calculus for the sum of noncommuting operators[END_REF]; here different techniques like estimates of the Green function, the theory of sum of operators or blowing up can be used.

∂ t u -∆u = h, in Ω × ] -π, π[, u = 0, on ∂Ω × [-π, π], u(•, -π) = u(•, π), in Ω,
According to the approach of [START_REF] Grisvard | Singular behavior of elliptic problems in non hilbertian Sobolev spaces[END_REF], the study of the linear heat equation in non-hilbertian Sobolev spaces can be performed with the help of the theory of sums of operators. Hence the goal of this paper is to make this analysis in L p (-π, π; L p µ (Ω)) for a large range of values of µ and p. Our results extend the ones from [START_REF] Grisvard | Edge behavior of the solution of an elliptic problem[END_REF][START_REF] Grisvard | Singular behavior of elliptic problems in non hilbertian Sobolev spaces[END_REF] to the L p µ (Ω) setting. This theory also requires, in a first step, to obtain uniform estimates of the solution of the Helmholtz equation

-∆u + zu = g in Ω, u = 0, on ∂Ω, (1.2)
where z is a complex number. This was performed in the companion paper [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF].

For the sake of simplicity we have restricted ourselves to two-dimensional domains Ω. The results of this paper can be easily extended to the case of domains with conical points.

The paper is organized as follows: In section 2 we recall some results on the sums of operators in Banach spaces of Da Prato-Grisvard [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] and of Dore-Venni [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF]; we also state some basic results from [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF] used later on. In section 3 we apply the approach of Da Prato-Grisvard to obtain a decomposition but with non-optimal regularity informations. Section 4 is devoted to the proof of the regularity of (∂ t -∆)S, where S is the singular part of the solution obtained before. The use of the approach of Dore-Venni and the results of section 4 allows to get the optimal regularity result obtained in section 5.

In the whole paper the notation a b means the existence of a positive constant C, which is independent of the quantities a, b (and eventually the above parameter z) under consideration such that a ≤ Cb.

Preliminary results

Results on the sums of operators in Banach spaces, such as the result of G. Da Prato and P. Grisvard [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] and of G. Dore and A. Venni [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF], can be fruitfully used to prove the singular behaviour of elliptic problems in non-Hilbertian Sobolev spaces as in [START_REF] Grisvard | Singular behavior of elliptic problems in non hilbertian Sobolev spaces[END_REF]. Let us recall these results. For an operator C we denote by σ(C) and ρ(C) respectively its spectrum and its resolvent set.

First strategy

Assumptions on A and B: (H 1 ) There exist M ≥ 0, R ≥ 0, θ A ∈ ]0, π], θ B ∈ ]0, π] such that

θ A + θ B > π, S A := {λ | |λ| ≥ R, | arg λ| ≤ θ A } ⊂ ρ(-A), S B := {λ | |λ| ≥ R, | arg λ| ≤ θ B } ⊂ ρ(-B),
and, for all λ ∈ S A and all µ ∈ S B ,

(A + λ I) -1 ≤ M |λ| , (B + µ I) -1 ≤ M |µ| ; (H 2 ) σ(-A) ∩ σ(B) = ∅;
(H 3 ) The resolvent of A and B commute, i.e., for every λ ∈ ρ(-A) and every µ ∈ ρ(-B),

(A + λ I) -1 (B + µ I) -1 = (B + µ I) -1 (A + λ I) -1 .
Theorem 2.1. [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelles[END_REF] Under assumptions (H 1 ), (H 2 ) and (H 3 ), the operator L has an invertible closure. The inverse of L is obtained as the integral

( L) -1 = 1 2iπ γ (A + λ I) -1 (λ I -B) -1 dλ,
where γ is a path which separates σ(-A) and σ(B) and joins ∞e -iθγ to ∞e iθγ where θ γ is chosen so that πθ B < θ γ < θ A .

Second strategy

Assumptions on A, B and E:

(H 4 ) E is a U.M.D. space; (H 5 ) ] -∞, 0] ⊂ ρ(A) ∩ ρ(B)
and there exists M ≥ 0 such that, for every t ≥ 0,

(A + t I) -1 ≤ M t + 1 , (B + t I) -1 ≤ M t + 1 ;
This allows to define the complex power of A and B by setting, for ℜ(z) < 0,

A z = - sin(πz) π +∞ 0 t z (A + t I) -1 dt.
This definition can be extended to ℜ(z) = 0 by taking limits when they exist.

(H 6 ) For every s ∈ R, the complex power A is and B is exist and are bounded operators. In addition there exist K > 0, τ A > 0, τ B > 0 such that

τ A + τ B < π,
and, for all s ∈ R,

A is ≤ Ke |s|τ A , B is ≤ Ke |s|τ B .
Theorem 2.2. [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF] Under assumptions (H 3 ), (H 4 ), (H 5 ) and (H 6 ), the operator L is invertible.

Results on the Helmholtz equation

In this paper, we work with a polygonal domain Ω of R 2 with a Lipschitz boundary ∂Ω, in the following sense.

Definition 2.2. Let Ω be a bounded domain of R 2 . We say that Ω is a polygonal domain if its boundary is the union of a finite number of line segments Γj , j ∈ {1, . . . , J} (Γ j being supposed to be open). Hence we do not assume that Ω is a Lipschitz domain, that is we include the presence of cracks.

Denote by S j , j = 1, • • • , J the vertices of ∂Ω enumerated clockwise. Without loss of generality we may assume that B(S j , 1) ∩ Ω does not contain any other vertex of Ω. For j ∈ {1, 2, • • • , J}, let ψ j be the interior angle of Ω at the vertex S j , λ j = π ψ j and (r j , θ j ) the polar coordinates centered at S j such that

B(S j , 1) ∩ Ω = {(r j cos θ j , r j sin θ j ) | 0 < r j < 1, 0 < θ j < ψ j } =: D j . For µ = (µ j ) J j=1 , we define the spaces L p µ (Ω) = {f ∈ L p loc (Ω) | wf ∈ L p (Ω)} with w = 1 + J j=1 η j (r µ j j -1), (2.1)
where r j (x) is the distance from x to the vertex S j and η j ∈ D(R 2 ) are such that η j ≡ 1 in D j (1/2), η j ≡ 0 on Ω \ D j (1), where D j (r) is the truncated cone D j (r) = Ω ∩ B(S j , r).

The space L p µ (Ω) is a Banach space for the norm

f L p µ (Ω) = Ω |f (x)| p w p (x) dx 1/p . V k,p µ (Ω) is defined as the closure of C ∞ S (Ω) = {v ∈ C ∞ (Ω) | S j ∈ supp v} with respect to the norm u V k,p µ (Ω) =   |γ|≤k Ω |D γ u(x)| p w p (x) r (|γ|-k)p (x) dx   1/p .
We use the following notation for the semi-norm

|u| V k,p µ (Ω) =   |γ|=k Ω |D γ u(x)| p w p (x) r (|γ|-k)p (x) dx   1/p .
In H 1 0 (Ω) we will denote the norms in the following way

|u| 2 H 1 0 = Ω |∇u| 2 and u 2 H 1 0 = Ω (|∇u| 2 + |u| 2 ).
For µ and γ, we write µ > γ in case, for all j ∈ {1, . . . , J}, µ j > γ j .

Let us finish this subsection by stating two theorems obtained in [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF] that concern uniform regularity results for the Helmholtz equation in weighted Sobolev spaces.

Theorem 2.3. [8] Let R > 0, p ≥ 2 and Ω be a bounded polygonal domain of R 2 . Denote λ = (λ j ) 1≤j≤J .
Let µ >λ satisfies, for all j = 1, . . . , J,

µ j < 2p-2 p , if p > 2, µ j ≤ 1, if p = 2, 4(p -1)λ 2 j -µ 2 j p 2 > 0 (2.2)
and, for all k ∈ Z * and all j ∈ {1, 2, • • • , J}, 2 -2 pµ j = kλ j . Then, there exists θ A ∈ ] π 2 , π[ such that, for all g ∈ L p µ (Ω), all z ∈ π + ∪S A , with

π + = {z ∈ C | ℜ(z) ≥ 0}, S A = {z ∈ C | |z| ≥ R and | arg z| ≤ θ A }, the problem -∆u + zu = g, in Ω, u = 0, on ∂Ω, has a unique solution u ∈ H 1 0 (Ω). Moreover this solution is in D(∆ p, µ ) := {u ∈ H 1 0 (Ω) | ∆u ∈ L p µ (Ω)
} and admits the decomposition

u = u R + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j c λ ′ j (z) P j,λ ′ j (r √ z)e -r √ z r λ ′ j sin(λ ′ j θ), (2.3
)

with u R ∈ V 2,p µ (Ω), c λ ′ j (z) ∈ C and P j,λ ′ j (s) = l j,λ ′ j -1 i=0 s i i! with l j,λ ′ j > 2 -µ j - 2 p -λ ′ j . Moreover, the following inequalities are satisfied (a) |u R | V 2,p µ (Ω) + |u R | V 1,p µ-1 (Ω) + u R L p µ-2 (Ω) g L p µ (Ω) ; (b) |u R | V 2,p µ (Ω) + |z| 1/2 |u R | V 1,p µ (Ω) + |z| u R L p µ (Ω) g L p µ (Ω) ; (c) J j=1 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j |c λ ′ j (z)| (1 + |z| 1-1 p - µ j +λ ′ j 2 ) g L p µ (Ω) .
Theorem 2.4. [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF] Under the assumptions of Theorem 2.3, D(∆ p, µ ) ⊂ L p µ (Ω) and we have

(a) If z ∈ C satisfies ℜ(z) ≥ 0 then ℜ(z) u L p µ (Ω) ≤ g L p µ (Ω) ; (b) If z ∈ C satisfies | arg z| ≤ θ A then (1 + |z|) u L p µ (Ω) g L p µ (Ω)
. Theorem 2.3 can be rephrased as follows. The operator (-∆ + z I) -1 can be decomposed as

(-∆ + z I) -1 = R(z) + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j T λ ′ j (z) ⊗ ψλ ′ j ,z (2.4)
where we use the linear and continuous operators

R(z) : L p µ (Ω) → V 2,p µ (Ω) : g → u R , T λ ′ j (z) : L p µ (Ω) → C : g → c λ ′ j (z) = T λ ′ j (z), g and the function ψλ ′ j ,z (r, θ) = P j,λ ′ j (r √ z)e -r √ z r λ ′ j sin(λ ′ j θ). Recall that (T λ ′ j (z) ⊗ ψλ ′ j ,z )(g) = T λ ′ j (z), g ψλ ′ j ,z . Moreover, for all z ∈ π + ∪ S A , we have R(z) L p µ (Ω)→V 2,p µ (Ω) + |z| 1/2 R(z) L p µ (Ω)→V 1,p µ (Ω) + |z| R(z) L p µ (Ω)→L p µ (Ω)
1, (2.5) and

T λ ′ j (z) (L p µ (Ω)) ′ 1 1 + |z| (1-1 p )- µ j +λ ′ j 2 .
(2.6)

Application of the first strategy

Let us assume in the future that the assumptions of Theorem 2.3 are satisfied.

Consider the problem (1.1) with h ∈ L p (I; L p µ (Ω)) with I = ]π, π[. In that case, h admits the decomposition h(x, t) = g 1 (x) + g(x, t) with, for a.e. x ∈ Ω, π -π g(x, t) dt = 0,

g 1 ∈ L p µ (Ω) and g ∈ L p (I; L p µ (Ω)).
To obtain such a decomposition, we just have to define

g 1 (x) = 1 2π π -π h(x, s) ds.
Hence u is a solution of (1.1) if and only if u(x, t) = ū(x) + v(x, t) with ū solution of

-∆ū = g 1 (x), in Ω, u = 0, on ∂Ω, (3.1)
and v(x, t) solution of

∂ t v -∆v = g(x, t), in Ω × ] -π, π[, v = 0, on ∂Ω × [-π, π], v(•, -π) = v(•, π), in Ω, π -π v(x, t) dt = 0, for all x ∈ Ω. (3.2)
By [START_REF] Kufner | Some applications of weighted Sobolev spaces[END_REF] and as in [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF], ū admits the decomposition

ū = ūR + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j cλ ′ j r λ ′ j sin(λ ′ j θ) (3.3) with ūR ∈ V 2,p µ (Ω), ūR V 2,p µ (Ω) g 1 L p µ (Ω)
and

|c λ ′ j | g 1 L p µ (Ω) .
Hence we concentrate on (3.2).

September 2, 2010

We shall apply the First Strategy (Theorem 2.1) on the space

E = {h ∈ L p (I; L p µ (Ω)) | for a.e. x ∈ Ω, π -π h(x, t) dt = 0}.
In the future, we will use the index m to denote the fact that the functions h of the space satisfy, for a.e. x ∈ Ω, π -π h(x, t) dt = 0. In that way

E =: L p m (I; L p µ (Ω)
). We consider the operators

A : D(A) ⊂ E → E : u → -∆u, with D(A) = L p m (I; D(∆ p, µ )) where D(∆ p, µ ) = {u ∈ H 1 0 (Ω) | ∆u ∈ L p µ (Ω)}, and
B 0 : D(B 0 ) ⊂ E → E : u → ∂ t u, with D(B 0 ) = W 1,p 2π,m (I; L p µ (Ω)) = {u ∈ E | ∂ t u ∈ L p (I; L p µ (Ω)), u(•, -π) = u(•, π)}. Proposition 3.1.
Under the assumptions of Theorem 2.3, the operator A + B 0 has an inverse closure i.e., for all g ∈ L p m (I;

L p µ (Ω)), there exists a unique strong solution v ∈ L p m (I; L p µ (Ω)) of (A + B 0 )v = g i.e. there exists (v n ) n ⊂ D(A) ∩ D(B 0 ) such that v n → v and Av n + B 0 v n → g. Moreover we have v = 1 2πi γ (A + z I) -1 (z I -B 0 ) -1 g dz, (3.4)
with γ : R → C defined for example by

γ(s) = |s| e -i( π 2 +δ) , for s ≤ 0, = |s| e i( π 2 +δ) , for s > 0, with δ ∈ ]0, θ A -π
2 [ and θ A given by Theorem 2.3. Proof. Observe that by Theorem 2.4, we have D(A) ⊂ E and, for all λ > 0,

(A + λI) -1 ≤ 1 λ . By [21, Thm I-4.2], this implies that -A is dissipative. As E is reflexive and R(I + A) = E, we have by [21, Thm I-4.6] that D(A) is dense in E.
Hence by Lumer-Phillips and Hille-Yosida Theorems, A is closed. It is easy to observe also that σ(

-A) = {-ν k | k ∈ N} where (ν k ) k is the strictly increasing sequence of eigenvalues of -∆ in H 1 0 (Ω). In particular ν 1 > 0 and lim k→∞ ν k = +∞. September 2, 2010
Concerning B 0 it is easy to observe that D(B 0 ) is a dense subset of E and that B 0 is closed. Moreover a simple calculation proves that σ(B 0 ) = iZ * and therefore ρ(

-B 0 ) ⊃ {λ ∈ C | ℜλ > 0}. Let us show that for all θ B < π 2 , there exists M ≥ 0 such that, for all µ ∈ S B 0 = {µ ∈ C | | arg(µ)| ≤ θ B }, (B 0 + µ I) -1 ≤ M |µ| . To this aim, it is enough to prove that if u is a solution of ∂ t u(x, t) + µ u(x, t) = f (x, t), in Ω × ] -π, π[, u(•, -π) = u(•, π), in Ω, π -π u(x, t) dt = 0, (3.5)
then |µ| u L p (I;L p µ (Ω)) f L p (I;L p µ (Ω))
. Multiplying the equation in (3.5) by v := w p |u| p-2 ū and integrating, we obtain

Ω π -π ∂ t u v dtdx + µ Ω π -π w p |u| p dtdx = Ω π -π w p f |u| p-2 ū dtdx. (3.6)
Observe that, by periodicity,

Ω π -π ∂ t u v dtdx = - Ω π -π ∂ t v u dtdx.
Moreover by [START_REF] Adeyeye | Generation of analytic semi-group in L p (Ω) by the Laplace operator[END_REF], we have

Ω π -π ∂ t u v dtdx = - p 2 Ω π -π w p |u| p-2 u ∂ t u dtdx - p -2 2 Ω π -π w p |u| p-4 u 2 u ∂ t u dtdx = - p 2 Ω π -π v ∂ t u dtdx - p -2 2 Ω π -π w p |u| p-2 u ∂ t u dtdx, i.e. p 2 Ω π -π v∂ t u dtdx + Ω π -π v∂ t u dtdx = 0.
Hence taking the real part of (3.6) gives

ℜ(µ) u L p (I;L p µ (Ω)) ≤ f L p (I;L p µ (Ω)) . As | arg(µ)| ≤ θ B < π 2 , we have |ℑ(µ)| ℜ(µ) and hence |µ| u L p (I;L p µ (Ω)) f L p (I;L p µ (Ω)) . (3.7)
We conclude that (H 1 ) is satisfied with θ A given by Theorem 2.3 and

θ B = π 2 -δ B with 0 < δ B < δ < θ A -π 2 .
It remains to verify (H 3 ). This can be easily deduced from the fact that the variables are separate in these two operators.

Hence we can apply Theorem 2.1 to conclude. Remark 3.1 Observe that, multiplying the equation

∂ t u = f -µu,
by |∂ t u| p-2 ∂ t u, integrating and using the inequality (3.7), we obtain also

(1 + |µ|) u L p (I;L p µ (Ω)) f L p (I;L p µ (Ω)) . (3.8)
As it is clear that, for each t, we have

[(A + z I) -1 h](t) = (-∆ + z I) -1 (h(t)),
we can use the decomposition (2.4) and rewrite (3.4) as

v = v R + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j v λ ′ j (3.9) with v R (x, t) = 1 2πi γ R(z)(z I -B 0 ) -1 g dz v λ ′ j (x, t) = 1 2πi γ T λ ′ j (z), (z I -B 0 ) -1 g ψλ ′ j ,z (x) dz = 1 2πi γ T λ ′ j (z), (z I -B 0 ) -1 g P j,λ ′ j (r √ z)e -r √ z r λ ′ j sin(λ ′ j θ) dz. (3.10) Proposition 3.2.
Under the assumptions of Theorem 2.3, let us denote

σ j := 1-1 p - µ j +λ ′ j 2 . For all s ∈ ]0, min(1-σ j , 1/p)[, for all g ∈ W s,p m (I, L p µ (Ω)), there exist qλ ′ j ∈ W s+σ j ,p m
(I) and Ẽλ ′ j such that v λ ′ j defined by (3.10) can be written as

v λ ′ j = ( Ẽλ ′ j * t qλ ′ j ) r λ ′ j sin(λ ′ j θ) = ( π -π Ẽλ ′ j (x, τ ) qλ ′ j (t -τ ) dτ ) r λ ′ j sin(λ ′ j θ).
Moreover we have

qλ ′ j (t) = 1 2πi γ T λ ′ j (z), (z I -B 0 ) -1 g dz, Ẽλ ′ j (x, t) = k∈Z * e ikt P j,λ ′ j (r √ ik) e -r √ ik ,
where we used the notations introduced at the end of Section 2, and the operator

U 0 : W s,p m (I, L p µ (Ω)) → W s+σ j ,p m (I) : g → qλ ′ j is continuous.
Remark 3.2 Observe that, by the domain of summation in (3.9), we have σ j > 0 and the condition µ j > -λ j implies that σ j < 1.

Proof. First observe that v λ ′ j ∈ L p ([-π, π]; L p µ (Ω)) and hence we can take its partial Fourier series in t.

Step 1: For all f ∈ L p µ (Ω), the application

C → C : z → T λ ′ j (z), f is holomorphic on A := {z ∈ C | | arg(z)| < θ A } and continuous on A. In fact the problem -∆u + zu = f, in Ω, u = 0, on ∂Ω, (3.11)
is equivalent to

-∆u = f -zu, in Ω, u = 0, on ∂Ω. (3.12)
By [START_REF] Kufner | Some applications of weighted Sobolev spaces[END_REF] as in [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF], we know that the solution u of this second problem admits the decomposition

u = u R + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j d λ ′ j r λ ′ j sin(λ ′ j θ) with d λ ′ j = d λ ′ j (f -zu) = Ω (f -zu)w λ ′ j
and w λ ′ j independent of z. Moreover we have

T λ ′ j (z), f = d λ ′ j (f -zu).
Hence it remains to consider the regularity of u with respect to z.

Let us consider the operator

S : C → L(L p µ (Ω), L p µ (Ω)) : z → (-∆ + z I) -1 . Observe that S(z)f -S(z + h)f = -h S(z) S(z + h) f
and hence

T λ ′ j (z + h), f -T λ ′ j (z), f h = 1 h Ω w λ ′ j (f -(z + h)S(z + h)f ) - Ω w λ ′ j (f -zS(z)f ) = 1 h z Ω w λ ′ j (S(z)f -S(z + h)f ) -h Ω w λ ′ j S(z + h)f = -z Ω w λ ′ j S(z) S(z + h) f - Ω w λ ′ j S(z + h)f.
By Theorem 2.4 we have C > 0 such that, for all f ∈ L p µ (Ω), z ∈ A and for h ∈ C small enough,

S(z + h)f -S(z)f L p µ (Ω) = |h| S(z) S(z + h) f L p µ (Ω) ≤ C |h| (1 + |z|) (1 + |z + h|) f L p µ (Ω) .
Hence we conclude that, for z ∈ A,

lim h→0 T λ ′ j (z + h), f -T λ ′ j (z), f h = -z Ω w λ ′ j S(z) 2 f - Ω w λ ′ j S(z)f,
and, for all f ∈ L p µ (Ω), the application C → C : z → T λ ′ j (z), f is holomorphic on A. In a similar way, we prove the continuity of this application on A.

Step 2: The Fourier series coefficient in

t of v λ ′ j (x, t) satisfies vλ ′ j (x, k) = -T λ ′ j (ik), ĝ(•, k) ψλ ′ j ,ik (x)
. By definition of the Fourier series in t and applying Fubini's theorem we obtain

vλ ′ j (x, k) = π -π e -ikt v λ ′ j (x, t) dt = 1 2πi γ T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz. For R > k, let us consider the domain D R bounded by γ 1,R : [-1, 1] → C defined by γ 1,R (s) = Re i( π 2 +δ)s and γ 2,R := -γ| [-R,R] and, for ǫ > 0 its subdomain D R,ǫ , bounded by γ 1,R and γ 2,R,ǫ := -γ| [-R,R] + ǫ.
As, for ǫ > 0, the function

z → T λ ′ j (z), ĝ(•, k) ψλ ′ j ,z (x)
is holomorphic in an open domain containing D R,ǫ then, by the Cauchy formula,

1 2πi ∂D R,ǫ T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz = T λ ′ j (ik), ĝ(•, k) ψλ ′ j ,ik (x).
As this fonction of ǫ is continuous for ǫ ∈ [0, ǫ 0 [ and constant, we deduce

1 2πi γ 1,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz + γ 2,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz = T λ ′ j (ik), ĝ(•, k) ψλ ′ j ,ik (x).
Let us prove that lim

R→+∞ γ 1,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz = 0, and γ T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz = -lim R→+∞ γ 2,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz.
Observe that by Hölder's inequality, we have

ĝ(•, k) p L p µ (Ω) = Ω w p (x) | π -π e -ikt g(x, t) dt| p dx Ω w p (x) π -π |g(x, t)| p dt dx g p L p (I;L p µ (Ω)) .
Hence by (2.6) we have

T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) |z| µ j +λ ′ j 2 -(1-1 p ) |z| -|k| |P j,λ ′ j (r √ z)e -r √ z | |r λ ′ j sin(λ ′ j θ)| g L p (I;L p µ (Ω)) , with µ j +λ ′ j 2 -(1 -1 p ) = -σ j < 0. This implies γ 1,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz 1 R σ j 1 -1 |P j,λ ′ j (r Re i( π 2 +δ)s ) e -r √ Re i( π 2 +δ)s | |r λ ′ j sin(λ ′ j θ)| ds g L p (I;L p µ (Ω)) ,
from which we deduce that lim

R→+∞ γ 1,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz = 0.
The argument to prove

γ T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz = -lim R→+∞ γ 2,R T λ ′ j (z), ĝ(•, k) z -ik ψλ ′ j ,z (x) dz, is similar.
Step 2 is then proved.

Step 3: The operator U 0 : W s,p m (I;

L p µ (Ω)) → W s+σ j ,p m (I) : g → qλ ′ j with qλ ′ j (t) = 1 2πi γ T λ ′ j (z), (zI -B 0 ) -1 g dz is continuous
. By the results of [START_REF] Grisvard | Equations différentielles abstraites[END_REF], as 0 < s < 1/p, we know that

W s,p m (I; L p µ (Ω)) = {g ∈ E | ∞ 0 ρ sp B 0 (B 0 -ρe ±i( π 2 +δ) I) -1 g p E dρ ρ < ∞}.
We have a similar characterization of W s+σ,p m (I) by considering the operator

N : D(N ) ⊂ E 1 → E 1 : u → ∂ t u with E 1 = L p m (I), D(N ) = {u ∈ E 1 | ∂ t u ∈ L p (I), u(-π) = u(π)}. Hence if s + σ j < 1/p then W s+σ j ,p m (I) = {g ∈ E 1 | ∞ 0 τ (s+σ)p N (N + τ I) -1 g p L p (I) dτ τ < ∞}, if s + σ j > 1/p then W s+σ j ,p 2π,m (I) = {g ∈ E 1 | ∞ 0 τ (s+σ)p N (N + τ I) -1 g p L p (I) dτ τ < ∞},
Claim 1: For τ ≥ 0, we have

N (N + τ I) -1 qλ ′ j = 1 2πi γ T λ ′ j (z), B 0 (zI -B 0 ) -1 g dz z + τ . (3.13)
First observe that

N (N + τ I) -1 qλ ′ j = 1 2πi γ T λ ′ j (z), B 0 (B 0 + τ I) -1 (zI -B 0 ) -1 g dz. (3.14)
Let us take the Fourier coefficients in t of

1 2πi γ T λ ′ j (z), B 0 (B 0 + τ I) -1 (zI -B 0 ) -1 g dz.
By Cauchy theorem, we obtain

1 2πi γ T λ ′ j (z), ik (ik + τ )(z -ik) ĝ(•, k) dz = -T λ ′ j (ik), ĝ(•, k) ik ik + τ .
In the same way, if we take the Fourier coefficients in t of

1 2πi γ T λ ′ j (z), B 0 (zI -B 0 ) -1 g dz z + τ ,
by Cauchy theorem, we obtain

1 2πi γ T λ ′ j (z), ik z -ik ĝ(•, k) dz z + τ = 1 2πi γ T λ ′ j (z), ( ik τ + ik 1 z -ik - ik τ + ik 1 z + τ )ĝ(•, k) dz = 1 2πi γ T λ ′ j (z), ik τ + ik 1 z -ik ĝ(•, k) dz = -T λ ′ j (ik), ĝ(•, k) ik τ + ik as T λ ′ j (z), ĝ(•,k)
z+τ is holomorphic on the right of γ. As the Fourier coefficients of the two functions coincide, the two functions are equal.

Claim 2: For 0 < s < min(1 -σ j , 1/p), the operator U 0 : W s,p m (I; L p µ (Ω)) → W s+σ j ,p m (I) : g → qλ ′ j is continuous. As 0 < s < 1/p, for g ∈ W s,p m (I; L p µ (Ω)) we have B 0 (B 0 -ρe i( π 2 +δ) I) -1 g E = η(ρ), with ∞ 0 ρ sp |η(ρ)| p dρ ρ < ∞.
By (3.13), denoting θ 0 = π 2 + δ, we have

N (N + τ I) -1 qλ ′ j = 1 2πi +∞ 0 T λ ′ j (ρe iθ 0 ), B 0 (ρe iθ 0 I -B 0 ) -1 g e iθ 0 dρ ρe iθ 0 + τ - 1 2πi +∞ 0 T λ ′ j (ρe -iθ 0 ), B 0 (ρe -iθ 0 I -B 0 ) -1 g e -iθ 0 dρ ρe -iθ 0 + τ ,
and hence

τ (s+σ j )p N (N + τ I) -1 qλ ′ j p L p (I) = τ (s+σ j )p π -π 1 2πi +∞ 0 T λ ′ j (ρe iθ 0 ), B 0 (ρe iθ 0 I -B 0 ) -1 g e iθ 0 dρ ρe iθ 0 + τ - 1 2πi +∞ 0 T λ ′ j (ρe -iθ 0 ), B 0 (ρe -iθ 0 I -B 0 ) -1 g e -iθ 0 dρ ρe -iθ 0 + τ p dt.

Using the inequality

+∞ 0 f L p (I) ≤ +∞ 0 f L p (I) , we have τ (s+σ j )p N (N + τ I) -1 qλ ′ j p L p (I) τ (s+σ j )p    +∞ 0   π -π T λ ′ j (ρe iθ 0 ), B 0 (ρe iθ 0 I -B 0 ) -1 g |ρe iθ 0 + τ | p dt   1/p dρ    p +τ (s+σ j )p    +∞ 0   π -π T λ ′ j (ρe -iθ 0 ), B 0 (ρe -iθ 0 I -B 0 ) -1 g |ρe -iθ 0 + τ | p dt   1/p dρ    p .
As the two terms behave in the same way, let us consider only the first term of the sum. We have, using (2.6),

τ (s+σ j )p    +∞ 0   π -π T λ ′ j (ρe iθ 0 ), B 0 (ρe iθ 0 I -B 0 ) -1 g |ρe iθ 0 + τ | p dt   1/p dρ    p τ (s+σ j )p +∞ 0 π -π ρ µ j +λ ′ j 2 -1+ 1 p B 0 (ρe iθ 0 I -B 0 ) -1 g L p µ (Ω) p dt 1/p dρ |ρe iθ 0 + τ | p τ (s+σ j )p +∞ 0 ρ µ j +λ ′ j 2 -1+ 1 p B 0 (ρe iθ 0 I -B 0 ) -1 g L p (I,L p µ (Ω)) dρ |ρe iθ 0 + τ | p τ (s+σ j )p +∞ 0 ρ µ j +λ ′ j 2 -1+ 1 p η(ρ) 1 |ρe iθ 0 + τ | dρ p   +∞ 0 ρ s η(ρ) τ s+σ j ρ µ j +λ ′ j 2 -1+ 1 p -s |ρe iθ 0 + τ | dρ   p .
As σ j = -

µ j +λ ′ j 2 + 1 -1 p we obtain τ (s+σ j )p N (N + τ I) -1 qλ ′ j p L p (I) +∞ 0 ρ s η(ρ) τ s+σ j |ρ| -(s+σ j ) |ρe iθ 0 + τ | dρ p = +∞ 0 ρ s η(ρ) (τ /ρ) s+σ j |e iθ 0 + τ ρ | dρ ρ p .
This is a multiplicative convolution and we have, by Young inequality, that qλ ′ j ∈ W s+σ j ,p (I) if

+∞ 0 ξ s+σ j |e iθ 0 + ξ| dξ ξ < ∞,
which is true as 0 < s + σ j < 1.

Conclusion. By

Step 2, we have

vλ ′ j (x, k) = -T λ ′ j (ik), ĝ(•, k) ψλ ′ j ,ik (x) = -T λ ′ j (ik), ĝ(•, k) P j,λ ′ j (r √ ik)e -r √ ik r λ ′ j sin(λ ′ j θ). (3.15) Let qλ ′ j (t) = 1 2πi γ T λ ′ j (z), (z I -B 0 ) -1 g dz.
Applying again the Cauchy theorem as above, we see that its Fourier series coefficient in t is given by qλ

′ j (k) = 1 2πi γ T λ ′ j (z), ĝ(•, k) z -ik dz = -T λ ′ j (ik), ĝ(•, k) .

Moreover the function Ẽλ

′ j (x, t) = k∈Z * e ikt P j,λ ′ j (r √ ik) e -r √ ik is such that, for all r > 0, Ẽλ ′ j (r, •) ∈ L 2 (-π, π) and even Ẽλ ′ j (r, •) ∈ C ∞ ([-π, π]). Hence we deduce from (3.15) that v λ ′ j (x, t) = ( Ẽλ ′ j * t qλ ′ j ) r λ ′ j sin(λ ′ j θ),
which allows to conclude.

Let us go back to the problem (1.1).

Proposition 3.3. Under the assumptions of Theorem 2.3, let σ j = 1 -1 p -

µ j +λ ′ j 2 .
Then, for all s ∈ ]0, min(1σ j , 1/p)[ and for all h ∈ W s,p (I, L p µ (Ω)), the problem (1.1) has a unique strong solution u with

u = u R + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j u λ ′ j
and, with the notations of Propositions 3.1, 3.2 and of (3.3),

u R (x, t) = 1 2πi γ R(z)(z I -B 0 ) -1 (h - 1 2π π -π h(•, s) ds) dz + ūR (x) u λ ′ j (x, t) = (E λ ′ j * t q λ ′ j ) r λ ′ j sin(λ ′ j θ) with q λ ′ j ∈ W s+σ j ,p (I) and E λ ′ j verifying q λ ′ j (t) = qλ ′ j (t) + cλ ′ j = 1 2πi γ T λ ′ j (z), (z I -B 0 ) -1 (h - 1 2π π -π h(•, s) ds) dz + cλ ′ j , E λ ′ j (x, t) = Ẽλ ′ j (x, t) + 1 2π = k∈Z * e ikt P j,λ ′ j (r √ ik) e -r √ ik + 1 2π .
Moreover, the operator

U : W s,p (I, L p µ (Ω)) → W s+σ j ,p (I) : h → q λ ′ j is continuous.
Proof. By the previous results it is enough to prove that

(E λ ′ j * t q λ ′ j ) r λ ′ j sin(λ ′ j θ) = v λ ′ j (x, t) + cλ ′ j .
Observe that, as

π -π qλ ′ j (s) ds = 0 and π -π
Ẽλ ′ j (r, ts) ds = 0, we have

(E λ ′ j * t q λ ′ j )(x, t) = π -π E λ ′ j (r, t -s) q λ ′ j (s) ds = π -π Ẽλ ′ j (r, t -s) q λ ′ j (s) ds + 1 2π π -π q λ ′ j (s) ds = π -π Ẽλ ′ j (r, t -s) q λ ′ j (s) ds + cλ ′ j = π -π Ẽλ ′ j (r, t -s) qλ ′ j (s) ds + cλ ′ j = ( Ẽλ ′ j * t q λ ′ j )(x, t) + cλ ′ j = v λ ′ j (x, t) + cλ ′ j . The result follows.
In the next result we extend Proposition 3.3 to h ∈ L p (I, L p µ (Ω)).

Theorem 3.4. Under the assumptions of Theorem 2.3, let

σ j = 1-1 p - µ j +λ ′ j 2 .
Then, for all h ∈ L p (I, L p µ (Ω)), the problem (1.1) has a unique strong solution u with

u = u R + J j=1 η j 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j u λ ′ j
and, with the notations of Propositions 3.1, 3.2 and of (3.3),

u R (x, t) = 1 2πi γ R(z)(z I -B 0 ) -1 (h - 1 2π π -π h(•, s) ds) dz + ūR (x) u λ ′ j (x, t) = (E λ ′ j * t q λ ′ j ) r λ ′ j sin(λ ′ j θ)
with q λ ′ j ∈ W σ j ,p (I) and E λ ′ j verifying

q λ ′ j (t) = qλ ′ j (t) + cλ ′ j = 1 2πi γ T λ ′ j (z), (z I -B 0 ) -1 (h - 1 2π π -π h(•, s) ds) dz + cλ ′ j , E λ ′ j (x, t) = Ẽλ ′ j (x, t) + 1 2π = k∈Z * e ikt P j,λ ′ j (r √ ik) e -r √ ik + 1 2π .
Moreover the mapping L p (I, L p µ (Ω)) → W σ j ,p (I) : h → q λ ′ j is continuous. Proof. We already know by Proposition 3.3 that, for 0 < s < min(1-σ j , 1/p), the operator U which maps h to q λ ′ j is continuous from W s,p (I; L p µ (Ω)) to W s+σ j ,p (I). We prove that U is also continuous from W s-1,p (I; L p µ (Ω)) to W s-1+σ j ,p (I), which allows to conclude by interpolation.

Claim 1: For 0 < s < min(1σ j , 1/p), the operator U : W s-1,p (I; L p µ (Ω)) → W s-1+σ j ,p (I) : h → q λ ′ j is continuous. We have

h ∈ W s-1,p (I; L p µ (Ω)) ⇔ ∃h 0 , h 1 ∈ W s,p (I; L p µ (Ω)), h = h 0 + ∂ ∂t h 1 .
Hence we define

U : W s-1,p (I; L p µ (Ω)) → W s-1+σ j ,p (I) by U h = U h 0 + ∂ ∂t U h 1 , where h = h 0 + ∂ ∂t h 1 with h 0 , h 1 ∈ W s,p (I; L p µ (Ω)
). Let us show that U is well defined. Assume that h admits a second decomposition h = h0 + ∂ ∂t h1 with h0 , h1 ∈ W s,p (I; L p µ (Ω)). In that case we have

h 0 -h0 = ∂ ∂t ( h1 -h 1 ),
and hence w := h1h 1 ∈ W s+1,p (I; L p µ (Ω)). As w ∈ W s+1,p (I; L p µ (Ω)) we have that D τ w = w(t+τ )-w(t) τ → w ′ in W s,p (I; L p µ (Ω)) as τ → 0. As U : W s,p (I; L p µ (Ω)) → W s+σ j ,p (I) is linear and continuous we have

D τ U w = U D τ w → U w ′ as τ → 0,
from which we deduce that U w is derivable and

∂ ∂t U w = U ∂ ∂t w
and hence, if

h 0 + ∂ ∂t h 1 = h0 + ∂ ∂t h1 , we have U (h 0 -h0 ) = U ( ∂ ∂t ( h1 -h 1 )) = ∂ ∂t U ( h1 -h 1 )
which means

U h 0 + ∂ ∂t U h 1 = U h0 + ∂ ∂t U h1
i.e. U well defined.

It remains to prove that U is continuous. We have for

h = h 0 + ∂ ∂t h 1 with h 0 , h 1 ∈ W s,p (I; L p µ (Ω)), U h W s-1+σ j ,p (I) U h 0 W s+σ j ,p (I) + U h 1 W s+σ j ,p (I) h 0 W s,p (I;L p µ (Ω)) + h 1 W s,p (I;L p µ (Ω))
h W s-1,p (I;L p µ (Ω)) , which proves the Claim.

Claim 2: L p (I; L p µ (Ω)) ֒→ (W s,p (I; L p µ (Ω)), W s-1,p (I; L p µ (Ω))) s,p . By Fubini's theorem, we have

W s,p (I, L p µ (Ω)) = L p µ (Ω, W s,p (I)),
as well as

W 1-s,p 0 (I, L p µ (Ω)) = L p µ (Ω, W 1-s,p 0 (I)),
and then by duality

W s-1,p (I, L p µ (Ω)) = L p µ (Ω, W s-1,p (I)). 4 Regularity of q λ ′ j → ( ∂ ∂t -∆)(η j u λ ′ j )
In order to consider the regularity of u R we observe that u R satisfies

∂ t u R + u R -∆u R = h - J j=1 0<λ ′ j <2-2 p -µ j ∃k∈N,λ ′ j =kλ j (∂ t (η j u λ ′ j -∆(η j u λ ′ j )) + u R
Hence we need informations on the regularity of ∂ t (η j u λ ′ j ) -∆(η j u λ ′ j ). This is the aim of this section. admits the decomposition

H(r, t) = H 1 (r, t) + H 2 (r, t), (4.2)
with

|H 1 (r, t)| 1, (4.3)
and

|H 2 (r, t)| 1 (r 2 + |t|) 3/2 . (4.4)
Moreover, for l ∈ N,

∂ 1+l ∂r 1+l H(r, t) = H (l) 1 (r, t) + H (l) 2 (r, t), with H (l) 1 and H (l) 2 satisfying |H
(l) 1 (r, t)| 1 (4.5)
and

|H (l) 2 (r, t)| (|t| + r 2 ) -(2+l/2) . (4.6)
Proof. Let us define the function

E (0) p (r, t) = k =0 e -|r| √ ik √ ik e ikt . (4.7)
We verify that

E (0) p ∈ L 2 (R × [-π, π]) as, using Parseval identity, +∞ -∞ k =0 |e -|r| √ ik | 2 |k| dr = 4 +∞ 0 k>0 e -2r √ k cos(π/4) |k| dr = 2 +∞ 0 k>0 e -s cos(π/4) |k| 3/2 ds = 2 k>0 1 |k| 3/2
+∞ 0 e -s cos(π/4) ds < +∞.

Considering the finite sum for |k| ≤ K and passing to the limit as K → +∞, we can take the Fourier transform in r of E (0) p using the fact that the Fourier transform is an isometry and

E (0) p ∈ L 2 (R × [-π, π]).
This gives

E (0) p (ξ, t) = k =0 1 √ ik e -|r| √ ik (ξ)e ikt .
As, for k = 0, we obtain

E (0) p (ξ, t) = 2 k =0 1 ξ 2 + ik e ikt . (4.8)
Let E be the elementary solution of the heat equation in R 2 i.e.

E(r, t) = M (t) √ 4πt e -r 2 4t , (4.9)
where

M (t) = 1, if t > 0, = 0, if t < 0.
Then we have that the Fourier transform of E in r is given by

E(ξ, t) = M (t) √ 4πt +∞ -∞ e -r 2 4t e -iξr dr = M (t) √ πt +∞ 0 e -r 2 4t cos(ξr) dr = M (t)e -ξ 2 t .
Consider now the function R 0 (r, t) which has as Fourier transform in r the function

R 0 (ξ, t) = - e -ξ 2 (t+π) 2 sinh(ξ 2 π) + 1 2ξ 2 π .
With these notations we have

( E -R 0 )(ξ, t) = 1 2π k∈Z β k e ikt ,
with

β k = 1 2 π -π e -ξ 2 (t+π) sinh(ξ 2 π) + 2M (t)e -ξ 2 t e -ikt dt - 1 2ξ 2 π π -π e -ikt dt.
Hence we have, for k = 0,

β k = 1 2 e -ξ 2 π sinh(ξ 2 π) π -π e -(ξ 2 +ik)t dt + 2 π 0 e -(ik+ξ 2 )t dt = -1 2(ξ 2 + ik) e -ξ 2 π sinh(ξ 2 π) (e -(ξ 2 +ik)π -e (ξ 2 +ik)π ) + 2(e -(ξ 2 +ik)π -1) = 1 ξ 2 + ik ,
and, in the same way, β 0 = 0.

Hence, we obtain

( E -R 0 )(ξ, t) = 1 2π k =0 1 ξ 2 + ik e ikt . (4.10)
By (4.8) and (4.10), we deduce that

E (0) p (ξ, t) = 4π( E(ξ, t) -R 0 (ξ, t))
and hence

E (0) p (r, t) = 4π(E(r, t) -R 0 (r, t)). Now observe that ∂ 2 ∂r 2 E (0) p (r, t) = k =0 √ ike -|r| √ ik e ikt -2δ 0 (r) k =0 e ikt = H(|r|, t) -2δ 0 (r)( k∈Z e ikt -1) = H(|r|, t) -2δ 0 (r)(2πδ 0 (t) -1) = H(|r|, t) -4πδ 0 (r)δ 0 (t) + 2δ 0 (r).
In the same way, we have

∂ 2 ∂r 2 (E -R 0 )(r, t) = ∂ 2 ∂r 2 E(r, t) - ∂ 2 ∂r 2 R 0 (r, t),
and, denoting R(ξ, t) = -e -ξ 2 (t+π) 2 sinh(ξ 2 π) , we obtain

∂ 2 ∂r 2 R 0 (ξ, t) = -ξ 2 R 0 (ξ, t) = -ξ 2 ( R(ξ, t) + 1 2ξ 2 π ) = -ξ 2 R(ξ, t) - 1 2π = -ξ 2 R(ξ, t) - 1 2π δ 0 (ξ) = ∂ 2 R ∂r 2 (ξ, t) - 1 2π δ 0 (ξ).
Hence we deduce

H(|r|, t) -4πδ 0 (r)δ 0 (t) = ∂ 2 ∂r 2 E (0) p (r, t) -2δ 0 (r) = 4π ∂ 2 E ∂r 2 (r, t) -4π( ∂ 2 R ∂r 2 (r, t) - 1 2π δ 0 (r)) -2δ 0 (r)
and therefore, for r > 0,

H(r, t) = 4π ∂ 2 E ∂r 2 (r, t) -4π ∂ 2 R ∂r 2 (r, t).
This suggests to decompose H(r, t) for r > 0 in the following way

H = H 1 + H 2 with H 1 = -4π ∂ 2 R ∂r 2
and

H 2 = 4π ∂ 2 E ∂r 2 . ( 4.11)
For H 1 we have

H 1 (ξ, t) = 4πξ 2 R(ξ, t) = -2π ξ 2 e -ξ 2 (t+π) sinh(ξ 2 π) .
Hence

H 1 (r, t) = 1 2π +∞ -∞ e irξ H 1 (ξ, t) dξ = - 1 π +∞ -∞ e irξ ξ 2 π sinh(ξ 2 π) e -ξ 2 (t+π) dξ
As t + π ≥ 0 we have e -ξ 2 (t+π) ≤ 1 and hence

|H 1 (r, t)| ≤ 1 π +∞ -∞ ξ 2 π sinh(ξ 2 π) dξ ≤ 1 π √ π +∞ -∞ x 2 sinh(x 2 ) dx, which implies (4.3) as +∞ -∞ x 2 sinh(x 2 ) dx < ∞.
Now let us consider the estimate (4.4). We have chosen

H 2 = 4π ∂ 2 E ∂r 2
, and hence we have,

H 2 (r, t) = -2π M (t) √ π t -3/2 ( 1 2 - r 2 4t ) e -r 2 4t .
This implies

|H 2 (r, t)| ≤ 2π t -3/2 √ π 1 2 - r 2 4t e -r 2 4t .
Recall that for all x ≥ 0 we have

|1 -x|e -x (1 + x) -3/2
and hence

|H 2 (r, t)| t -3/2 (1 + r 2 t ) -3/2 (t + r 2 ) -3/2 .
The result concerning the derivatives can be deduced easily by similar considerations. For what concerns H (l) 1 , we just have to observe that

H (l) 1 (r, t) = ∂ 1+l H 1 (r, t) ∂r 1+l = +∞ -∞
e irξ (iξ) 3+l sinh(ξ 2 π) e -ξ 2 (t+π) dξ and hence, as above, |H

1 (r, t)| 1. On the other hand, by recurrence, we can prove that, for some a i ∈ R,

H (l) 2 (r, t) = ∂ 1+l H 2 (r, t) ∂r 1+l = 4π ∂ 3+l E(r, t) ∂r 3+l
= 2π

M (t) √ π 1 √ t e -r 2 4t   [ 3+l 2 ] i=0 a i r 3+l-2i t 3+l-i   = 2π M (t) √ π 1 t 4+l 2 e -r 2 4t   [ 3+l 2 ] i=0 a i r 2 t 3+l 2 -i   ,
and we conclude as above.

Theorem 4.2. Under the assumptions of Theorem 3.4 and recalling that

σ j = 1 -1 p - µ j +λ ′ j 2 , the mapping q λ ′ j → ( ∂ ∂t -∆)(η j u λ ′ j ) is continuous from W σ j ,p (I) into L p (I; L p µ (Ω)). Proof.
Recall that, by Remark 3.2, 0 < σ j < 1.

Case 1: P j,λ ′ j ≡ 1 i.e. λ ′ j + µ j -1 + 2 p > 0. Let us take the Fourier series in

t of f = η j ( ∂ ∂t -∆)u λ ′ j . We obtain fk = η j (r) (( ∂ ∂t -∆)u λ ′ j ) k = η j (r) (ik I -∆) ûλ ′ j ,k = -c λ ′ j (ik) (2λ ′ j + 1) √ ik e -r √ ik r λ ′ j -1 sin(λ ′ j θ) η j (r), with c λ ′ j (ik) = T λ ′ j (ik), ĝ(•, k) = -q λ ′ j (k).
Let us consider the kernel H given by Lemma 4.1 i.e.

H(r, t) = k∈Z √ ik e -r √ ik e ikt = k∈Z H k (r) e ikt .
Hence, we have

f = (H * t q λ ′ j ) (2λ ′ j + 1) r λ ′ j -1 sin(λ ′ j θ) η j (r) = (2λ ′ j + 1) r λ ′ j -1 sin(λ ′ j θ) η j (r) π -π
H(r, s) q λ ′ j (ts) ds.

As 1 2π

π -π H(r, s) ds = H 0 (r) = 0, we have

f = (2λ ′ j + 1) r λ ′ j -1 sin(λ ′ j θ) η j (r) π -π
H(r, s) [q λ ′ j (ts)q λ ′ j (t)] ds. (4.12)

By (4.2) we define f = f 1 + f 2 with

f i = (2λ ′ j + 1) r λ ′ j -1 sin(λ ′ j θ) η j (r) π -π H i (r, s) [q λ ′ j (t -s) -q λ ′ j (t)] ds.
Step 1: Study of f 1 . By (4.3) we have

|f 1 (x, t)| r λ ′ j -1 η j (r) π -π |q λ ′ j (t -s) -q λ ′ j (t)| ds r λ ′ j -1 η j (r) { π -π |q λ ′ j (s)| ds + 2π |q λ ′ j (t)|} r λ ′ j -1 η j (r) { q λ ′ j W σ j ,p (I) + |q λ ′ j (t)|}. This implies that π -π Ω |f 1 (x, t)| p r µ j p r drdt ( 1 0 r (λ ′ j -1)p+µ j p+1 dr) 2π q λ ′ j p W σ j ,p (I) + π -π |q λ ′ j (t)| p dt q λ ′ j p W σ j ,p (I) 1 0
r (λ ′ j -1)p+µ j p+1 dr.

The integral 1 0 r (λ ′ j -1)p+µ j p+1 dr converges if (λ ′ j -1) + µ j + 2 p > 0 which is the condition to have P j,λ ′ j ≡ 1.

Step 2: Study of f 2 . By (4.4), we have By the assumption q λ ′ j ∈ W σ j ,p (I) we have

|f 2 (x, t)| r λ ′ j -1 η j (r) π -π |q λ ′ j (t -s) -q λ ′ j (t)| (r 2 + |s|) 3/2 ds from which we deduce π -π |f 2 (x, t)| p dt 1/p r λ ′ j -1 η j (r) π -π (r 2 + |s|) -3 2 ( π -π |q λ ′ j (t -s) -q λ ′ j (t)| p dt)
I 2 |q λ ′ j (x) -q λ ′ j (y)| p |x -y| 1+σ j p dxdy ≤ q λ ′ j p W σ j ,p (I) < ∞. (4.14)
Making the change of variables (x, y) = (ts, t), (4.14) becomes

I 2 |q λ ′ j (t -s) -q λ ′ j (t)| p |s| 1+σ j p dsdt ≤ q λ ′ j p W σ j ,p (I) . Denoting κ(s) = π -π |q λ ′ j (t -s) -q λ ′ j (t)| p |s| 1+σ j p dt 1/p , this estimates is equivalent to π -π |κ(s)| p ds ≤ q λ ′ j p W σ j ,p (I) . (4.15)
Going back to (4.13), we obtain

f 2 (x, •) L p (I) r λ ′ j -1 η j (r) π -π (r 2 + |s|) -3 2 |s| 1 p +σ j κ(s) ds, which implies π -π Ω |f 2 (x, t)| p r µ j p rdrdθdt ≤ 1 0 r (λ ′ j -1+µ j )p+2 π -π (r 2 + |s|) -3 2 |s| 1 p +σ j +1 κ(s) ds |s| p dr r .
(4.16) Observe that, by definition of σ j ,

r (λ ′ j -1+µ j )+ 2 p (r 2 + |s|) -3 2 |s| σ j +1 = r (λ ′ j -1+µ j )+ 2 p |s| σ j +1-3 2 ( r 2 |s| + 1) -3 2 = r (λ ′ j -1+µ j )+ 2 p |s| 2-1 p - µ j +λ ′ j 2 -3 2 ( r 2 |s| + 1) -3 2 = ( r 2 |s| ) (λ ′ j -1+µ j )+ 2 p 2 ( r 2 |s| + 1) -3 2 = k( r 2 |s| ) where k(τ ) = τ (λ ′ j -1+µ j )+ 2 p 2 (τ + 1) -3 2 .
With these notations, (4.16) becomes

f 2 p L p (I;L p µ (Ω)) ≤ 1 0 ( π -π k( r 2 |s| ) |s| 1 p κ(s) ds |s| ) p dr r . (4.17)
Making the change of variables r ′ = r 2 , (4.17) becomes

f 2 p L p (I;L p µ (Ω)) 1 0 ( π -π k( r ′ |s| ) |s| 1 p κ(s) ds |s| ) p dr ′ r ′ 1 0 ( π 0 k( r ′ |s| ) |s| 1 p κ(s) ds |s| ) p dr ′ r ′ . Let us define κ(s) = κ(s), if s ∈ ]0, π[, = 0, otherwise.
In that case

π 0 k( r ′ s ) s 1 p κ(s) ds s = ∞ 0 k( r ′ s ) s 1 p κ(s) ds s = (k * m (s 1 p κ))(r ′ )
where * m denotes the multiplicative convolution. Now observe that, by (4.15), we have It remains to verify that k ∈ L 1 (0, ∞; dr r ) i.e.

∞ 0 r µ j +λ ′ j -1+ 2 p 2 (r + 1) -3 2 dr r < ∞.
This is true as, by the summation domain λ ′ j + µ j + 2 p < 2 and as P j,λ

′ j ≡ 1, λ ′ j + µ j -1 + 2 p > 0. Case 2: deg(P j,λ ′ j ) = l j,λ ′ j -1 ≥ 1. In that case, we have fk = η j (r) (ikI -∆)û λ ′ j ,k = c λ ′ (ik) ik e -r √ ik r λ ′ j sin(λ ′ j θ) η j (r) P (r √ ik) (4.18)
with P a polynomial function of degree l j,λ ′ j -2 and l j,λ ′ j such that

l j,λ ′ j + λ ′ j + µ j -2 + 2 p > 0. ( 4

.19)

Let us introduce the kernel

H (l) (r, t) = k∈Z (ik) 1+l/2 e -r √ ik e ikt ,
where l = 0, 1, ...l j,λ ′ j -2. We see that

|H (l) | = ∂ 1+l ∂r 1+l H .
By Lemma 4.1, we have

H (l) = H (l) 1 + H (l) 2 , (4.20)
with

H (l)
1 and H (l)

2 satisfying (4.5) and (4.6). In the same way as before, we define

f (l) i = r λ ′ j +l sin(λ ′ j θ) η j (r) π -π H (l) i (r, s) [q λ ′ j (t -s) -q λ ′ j (t)] ds. and we have |f | l j,λ ′ j -2 l=0 (|f (l) 2 | + |f (l) 1 |), (4.21) with |f
(l) i (x, t)| = r λ ′ j +l η j (r) π -π |H (l) i (r, s)| |q λ ′ j (t -s) -q λ ′ j (t)| ds. (4.22)
Hence we have, as in the first case,

π -π Ω |f (l) 1 (r, t)| p r µ j p rdrdt q λ ′ j W σ j ,p (I) 1 0 r (λ ′ j +µ j +l)p+1 dr,
where the last integral converges as λ ′ j + µ j + l + 2 p > 0 which is true as σ j < 1 and l ≥ 0.

In the same way as before, we have

f (l) 2 p L p (I;L p µ (Ω)) 1 0 r (λ ′ j +l+µ j )p+2 π -π (r 2 + |s|) -(2+ l 2 ) |s| 1 p +σ j +1 κ(s) ds |s| p dr r 1 0 |k (l) * m (s 1/p κ))(r)| p dr r with k (l) (u) = u λ ′ j +l+µ j 2 + 1 p (u + 1) -(2+l/2) .
We conclude observing that, as 0 < σ j < 1, we have

∞ 0 k (l) (u) du u < ∞.
Conclusions. Now observe that

( ∂ ∂t -∆)(η j u λ ′ j ) = η j ( ∂ ∂t -∆)u λ ′ j -2 ∂η j ∂r ∂u λ ′ j ∂r -u λ ′ j ∆η j with ∂η j
∂r and ∆η j equals to zero on D j (1/2) ∪ (Ω \ D j (1)). Hence it is easy to deduce that

η j ( ∂ ∂t -∆)u λ ′ j -2
∂η j ∂r ∂u λ ′ j ∂r u λ ′ j ∆η j ∈ L p (I, L p µ (Ω)), which concludes the proof.

Application of the second strategy

Now we are able to consider the regularity of u R and to prove our main result.

Theorem 5.1. Let p ≥ 2, Ω be a bounded polygonal domain of R 2 and denote λ = (λ j ) 1≤j≤J . Let µ satisfies, for all j = 1, . . . , J, -λ j < µ j < 2p-2 p , 4(p -1)λ 2 jµ 2 j p 2 > 0 and, for all k ∈ Z * and all j ∈ {1, 2, • • • , J}, 2-2 p -µ j = kλ j and µ j +kλ j = 1. Let σ j = - The assumptions (H 3 ) and (H 5 ) can be verified as previously. The assumption (H 4 ) is satisfied by all L p µ (Ω) spaces (see for example [START_REF] Amann | Linear and quasilinear parabolic problems[END_REF]). It remains to verify (H 6 ). To this aim we will apply the following result of Coifman -Weiss (see [START_REF] Coifman | Transference methods in analysis[END_REF] or for example [START_REF] Amann | Linear and quasilinear parabolic problems[END_REF]).

If -A is the infinitesimal generator of a strongly continuous contraction semigroup in E which preserves the positivity then there exists K > 0 such that, for all s ∈ R,

A is ≤ K(1 + |s|) e
As f is real, the solution u is real. Let us decompose u = u +u -with u + = max(u, 0) and u -= max(-u, 0). Hence we have u + , u -∈ H 1 0 (Ω) and by (5.2) applied with w = u -, we obtain

Ω |∇u -| 2 + λ Ω |u -| 2 = - Ω f u -≤ 0.
Hence u -≡ 0 and u ≥ 0.

By [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] we have that if -A is the generator of T (t) then, for all f ∈ E, T By what we prove just before we have (λI + A) -1 f ≥ 0 and hence T (t)f ≥ 0. We then deduce that the semi-group preserves the positivity. Hence, there exists K > 0 such that, for all s ∈ R, A is L p (I,L p µ (Ω)) ≤ A is 1-θ L 2 (I,L 2 (Ω)) A is θ L q (I,L q ν (Ω))

≤ K θ (1 + |s|) θ e θ π 2 |s| .
Hence, for all such θ, for all ǫ > 0 there exists K(ǫ, θ) > 0 such that e -(1+λ)(π-s) e (1+λ)πe -(1+λ)π f (x, s) ds e -(1+λ)t + t -π e -(1+λ)(t-s) f (x, s) ds ≥ 0.

Fix a complex

 complex Banach space E and a pair of closed linear densely defined operators A : D(A) ⊂ E → E and B : D(B) ⊂ E → E. Hence we can define their sum L : D(L) := D(A) ∩ D(B) ⊂ E → E : x → Lx := Ax + Bx.

Definition 2 . 1 .

 21 The closure of L is defined by x ∈ D( L) and Lx = y if there exists a sequence (x n ) n ⊂ D(L), which satisfies x n → x and Lx n → y.A solution of Lx = y is called a strong solution of Lx = y.

Lemma 4 . 1 .

 41 The kernel H defined onR + × [-π, π] by H(r, t) = k∈Z √ ik e -r √ ik e ikt = k∈Z H k (r) e ikt (4.1)

  -iξr + e iξr ) dr = 2 √ ik ξ 2 + ik ,

  )| p ds ≤ q λ ′ j p W σ j ,p (I) .

µ j +λ ′ j 2 + 1

 21 -1 p , then, for every h ∈ L p (I; L p µ (Ω)), there exists a unique solution u ∈ L p (I;L p µ (Ω)) of ∂ t u -∆u = h(x, t), in Ω × ]π, π[, u = 0, on ∂Ω × [-π, π], u(•, -π) = u(•, π), in Ω.

andA

  : D(A) ⊂ E → E : u → -∆u, with D(A) = L p (I; D(∆ p, µ )), B : D(B) ⊂ E → E : u → ∂ t u + u, with D(B) = W 1,p 2π (I; L p µ (Ω)).

π 2

 2 |s| .For what concerns the operator A, we already know (see[8, proof of Corollary 2.14]) that -A generates a C 0 semigroup of contractions T (t). It remains to prove that T (t) preserves the positivity.Let f ∈ L p µ (Ω) with f ≥ 0, λ ∈ [0, +∞[ and u ∈ H 1 0 (Ω) be the solution of ∀w ∈

  t)f = lim λ→∞ e t(λ 2 (λI+A) -1 -λI) f = lim λ→∞ e -λt e tλ 2 (λI+A) A) -k f.

A

  is L p (I;L p µ (Ω)) ≤ K(1 + |s|) e π 2 |s| .As -A is symmetric on the Hilbert space L 2 (I, L 2 (Ω)), we have also (see for example[3, p. 164])A is L 2 (I,L 2 (Ω)) ≤ 1. Hence under the assumptions of Theorem 2.3, for θ ∈ ]0, 1[ close enough to 1 in such a way that ν = µ θ and q = 2pθ 2-p(1-θ) satisfy the assumptions of Theorem 2.3, we have, by [25, 1.18.7/Th 4],

+ 1 ,

 1 A is L p (I,L p µ (Ω)) ≤ K(ǫ, θ) e (θ π 2 +ǫ)|s| , from which we deduce the existence of τ A < π 2 such that A is L p (I,L p µ (Ω)) = 0(e τ A |s| ). For what concerns B, observe that σ(-B) = {-(ki + 1) | k ∈ Z} and hence σ(-B) ∩ [0, +∞[ = ∅. Moreover we have seen that R + ⊂ ρ(-B) and, for all λ ∈ R + , (λI + B) -1 ≤ 1 λ and hence, as in [8, proof of Corollary 2.14], we see that -B is the generator of a C 0 semigroup S(t) of contraction. Let us show that S(t) preserves the positivity. Consider the solution u ∈ D(B) of ∂ t u + u + λu = f ≥ 0, u(-π) = u(π), then u(x, t) = (B + λ I)

[START_REF] Solonnikov | L p -estimates for solutions of the heat equation in a dihedral angle[END_REF]

Hence we may write (W s,p (I, L p µ (Ω)), W s-1,p (I, L p µ (Ω))) s,p = (L p µ (Ω, W s,p (I)), L p µ (Ω, W s-1,p (I))) s,p , and applying [START_REF] Triebel | Interpolation theory, function spaces, differential operators[END_REF]Theorem 1.18.4,p.128], we deduce that (W s,p (I, L p µ (Ω)), W s-1,p (I, L p µ (Ω))) s,p = L p µ (Ω, (W s,p (I), W s-1,p (I)) s,p ).

Hence by [START_REF] Grisvard | Commutativité de deux foncteurs d'interpolation et applications[END_REF]Thm 6.2], we obtain (W s,p (I, L p µ (Ω)), W s-1,p (I, L p µ (Ω))) s,p = L p µ (Ω, 0,p (I)).

As Taibleson's results [START_REF] Taibleson | On the theory of Lipschitz spaces of distributions on euclidian n-space[END_REF] yield L p (I) ֒→ B 0,p (I), we have shown that

We conclude by Fubini's theorem that

Conclusion. By interpolation the application

As, for σ j ∈ N, (W s+σ j ,p (I), W s-1+σ j ,p (I)) s,p = B σ j ,p (I) = W σ j ,p (I), and, by Remark 3.2, σ j ∈ ]0, 1[, we have a continuous operator U : L p (I; L p µ (Ω)) → W σ j ,p (I) : h → q λ ′ j , and the result follows.

Moreover u admits the decomposition

Proof. Recall that, in the notations of the end of Section 2, we define

where u R is the regular part of the solution of

and, for all

1.

Hence by interpolation we have (see for example [START_REF] Adams | Sobolev spaces[END_REF]Thm 7.22]), for all θ ∈

In that case, by Theorem 3.4 and Remark 3.1

Hence, for all θ ∈ [0, 1[,

with the estimate

for some positive constant K(θ) that may depend on θ but not on h.

Let us show that

First observe that u R is a strong solution of

with, by the previous results, h R ∈ L p (I; L p µ (Ω)). Then we apply the second strategy with

As above we deduce that S(t) preserves the positivity. By the previous result of Coifman-Weiss, there exists K > 0 such that, for all s ∈ R,

Hence we obtain τ A < π/2 and τ B ∈ ]π/2, πτ A [ such that

As all the assumptions of the second strategy are satisfied, we have the existence of w R ∈ W 1,p 2π (I;

in Ω.

(

In fact, by Theorem 3.4, we know that u is a strong solution of

Moreover, as in Section 3, for every n, we have the decomposition

By Theorems 3.4 and 4.2, we have

. By the estimate (5.1) we see that u n,R → u R in E. Now observe that w R is a strong solution of (5.3) as w n,R = w R are such that w n,R ∈ D(A) ∩ D(B) and w n,R → w R .

Hence applying the first strategy to (5.3) with

we have by uniqueness of the strong solution that w R = u R and hence [START_REF] Kufner | Some applications of weighted Sobolev spaces[END_REF] as in [START_REF] Coster | Singular behavior of the solution of the Helmholtz equation in weighted L p -Sobolev spaces[END_REF], u admits the decomposition

Recall that by [START_REF] Coster | Lower and upper solutions for elliptic problems in nonsmooth domains[END_REF] we have that

as well as L p µ (Ω) → L p (Ω) : u → w u are continuous. Hence, if u ∈ (L p µ (Ω), V 2,p µ (Ω)) θ we have w u ∈ (L p (Ω), W 2,p (Ω)) θ . By [START_REF] Adams | Sobolev spaces[END_REF] we know (L p (Ω), W 2,p (Ω)) θ = W 2θ,p (Ω) and in particular r µ j u ∈ W 2θ,p (D j ).

By [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] we have if µ j + λ ′ j > 2θ -2 p , then r µ j +λ ′ j sin(λ ′ j θ) ∈ W 2θ,p (D j ), if µ j + λ ′ j ≤ 2θ -2 p and µ j + λ ′ j ∈ N, then r µ j +λ ′ j sin(λ ′ j θ) ∈ W 2θ,p (D j ). As µ j + λ ′ j < 2 -2 p , for θ close to 1, we have µ j + λ ′ j ≤ 2θ -2 p and hence u = u 1 ∈ V 2,p µ (Ω).