Regularity and Comparison Results in Grand Sobolev Spaces for Parabolic Equations with Measure Data

A. Fiorenza*

Dipartimento di Costruzioni e Metodi Matematici in Architettura Universitá di Napoli ”Federico II” via Monteoliveto, 3 I-80134 Napoli, Italy

                             fiorenza@unina.it

A. Mercaldo*

Dipartimento di Matematica e Applicazioni ”R. Caccioppoli” Universitá di Napoli ”Federico II”, Complesso Monte S. Angelo via Cintia, I-80126 Napoli, Italy

                       mercaldo@matna2.dma.unina.it

J.M. Rakotoson

Laboratoire d’Applications des mathématiques Teleport 2 Départment de Mathématiques Université de Poitiers B.P. 30179 86962 Futurscope chasseneuil cedex France

                       rako@matpts.univ-poitiers.fr

Abstract - We extend the results of [4, 5, 6, 2] on the uniqueness of solutions of parabolic equations. Our results give also some regularity results which complete the existence results made in [1, 7, 8].

1. INTRODUCTION

Consider Ω an open bounded set of ℝ^{N}, with smooth boundary and T > 0. We shall define the following spaces called Grand Lebesgue-Sobolev spaces:

L^{N)}(Ω) = \{v ∈ L^{1}(Ω) : sup_{0<ε<1/(2N)} (ε ∫_{Ω} |v|^{N−εN}dx)^{\frac{1}{N−εN}} < +∞\}

with norm

[v] = sup_{0<ε<1/(2N)} (ε ∫_{Ω} |v|^{N−εN}dx)^{\frac{1}{N−εN}}

*This work has been partially performed as a part of a National Research Project supported by M.U.R.S.T.

Typeset by AMS-TEX

We shall denote by

W_{0}^{1,N)}(Ω) = {v ∈ W_{0}^{1,1}(Ω) : |∇u| ∈ L^{N)}(Ω)}

As for the usual Sobolev spaces for time depending functions, we shall consider the following Grand Sobolev spaces:

L^{N)}(0, T; W_{0}^{1,N)}(Ω)), L^{N′)}(0, T; W^{−1,N′)}(Ω)), 1/N + 1/N′ = 1

where W^{−1,N′)}(Ω) denotes the dual space of W_{0}^{1,N)}(Ω).

We shall set Q = (0, T) × Ω, ∑_{T} (0, T) × ∂Ω. Consider a Charathéodory map â: Ω × ℝ^{n} → ℝ^{n} satisfying the following growth, for a.e. x ∈ Ω, ξ ∈ ℝ^{n}, ξ′ ∈ ℝ^{n}

[â(x, ξ) − â(x, ξ′)][ξ − ξ′] ≥ |ξ − ξ′|^{N}, α > 0
|â(x, ξ)| ≤ a_{0}|ξ|^{N_{1}} + a_{1}, a_{i} > 0, a_{i} ∈ ℝ.

Let μ ∈ L^{1}(0, T; M(Ω)), u_{0} ∈ M(Ω). We shall consider a solution u (when it exists) of:

P_{(u_{0},μ,â)} \{ \begin{matrix}∂_{t}u − div(â(x, ∇u)) = μ, in D′(Q) \\ u ∈ L^{N)}(0, T; W_{0}^{1,N)}(Ω)) \\ u(0) = u_{0} \end{matrix}

Existence of solution of P_{(u_{O},μ,â)} have been proven in [5], [6] in the usual Sobolev spaces. To garantee the existence in the Grand Sobolev spaces, we shall assume more regularity on u_{0} and μ (see Theorem 2.2 below).

In the detailed paper [1], we will complete the results that we will stated here for a more general structure.

2. MAIN RESULTS

Theorem 2.1 Let μ ∈ L^{N′)}(0, T; M(Ω)) and u a solution of P_{(u_{O},μ,â)}. Then

∂_{t}u ∈ L^{N′)}(0, T; W^{−1,N′)}(Ω)).

Moreover if u ∈ L^{q(N−1)}(0, T; W_{0}^{1,N)}(Ω)) and μ ∈ L^{q}(0, T; M(Ω)), 1 ≤ q ≤ +∞, then

∂_{t}u ∈ L^{q}(0, T; W^{−1,N′)}(Ω)).

The regularity of u can be obtained, for some solution u obtained by approximation. Indeed, we have:

Theorem 2.2 Suppose that there exists a sequence u_{n} ∈ L^{N}(0, T; W_{0}^{1,N}(Ω)), u_{0n}, μ_{n}, â_{n} such that u_{n} is a solution of P_{(u_{On},μ_{n},â_{n})} converging to u solution of

P_{(u_{O},μ,â)} in L^{1}(0, T; W_{0}^{1,1}(Ω)). If μ ∈ L^{N′)}(0, T; M(Ω)) (resp. L^{q}(0, T; M(Ω))) and

∂_{t}u_{n} remains in a bounded set of L^{N′)}(0, T; M(Ω)) (resp. L^{q}(0, T; M(Ω))), then u ∈ L^{N)}(0, T; W_{0}^{1,N)}(Ω)) (resp. u ∈ L^{q(N−1)}(0, T; W_{0}^{1,N)}(Ω)).

The sequence u_{n} do exist for instance for the operator N−Laplacian and u_{0} smooth.

A regularity of the difference of two solutions can be also obtained; for instance, one has:

Theorem 2.3 If (u, v) are two solutions of P_{(u_{O},μ,â)} and ∂_{t}u (resp. ∂_{t}v) is in

L^{1}(0, T; M(Ω)) + L^{N′}(0, T; W^{−1,N′}(Ω)), then T_{k}(u − v) ∈ L^{N}(0, T; W_{0}^{1,N}(Ω)) with T_{k}(σ) = \frac{1}{2}\{|k + σ| − |k − σ|\}, σ ∈ ℝ.

The following theorem that we will state is the following comparison theorem

Theorem 2.4 Let (u, v) be two solutions of P_{(u_{O},μ,â)} with u_{0} ∈ L^{1}(Ω). If ∂_{t}u

(risp. ∂_{t}v) is in L^{1}(Q) + L^{N′}(0, T; W^{−1,N′}(Ω)), then u = v.

Uniqueness theorem will be also given in [3]; for instance, we can show the following result:

Theorem 2.5 Assume that â(x, ξ) = A(x)ξ, with A a bounded coercive matrix. Then there is at most one solution u of P_{(u_{O},μ,â)} whenever u_{0} ∈ M(Ω), μ ∈ L^{1}(0, T; M(Ω)).

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