Regularity and uniqueness results

in Grand Sobolev Spaces

for parabolic equations with measure data

A. Fiorenza ^{1}“*”

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^{2}“*” Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Universitá di Napoli “Federico II”, via Cintia, I-80126 Napoli, Italy mercaldo@unina.it

J.M. Rakotoson Laboratoire d’Applications des Mathématiques Teleport 2 Département de Mathématiques Université de Poitiers B.P. 30179 86962 Futuroscope Chasseneuil cedex, France rako@mathlabo.univ-poitiers.fr

Abstract

In this paper some regularity and uniqueness results are obtained for parabolic equations whose prototype is ∂_{t}u − Δ_{N}u = μ, where the right-hand side μ is a Radon measure. The results are obtained in the framework of the so-called Grand Sobolev spaces, and represent an extension of earlier results on standard Sobolev spaces.

1 Introdution

In the present paper, announced in [8], we consider a class of parabolic problems whose prototipe is

\{ \begin{matrix}∂_{t}u − Δ_{N}u = μ & in D′(Q) \\ u = 0 & on ]0, T [×∂Ω \\ u(0) = u_{0} & in Ω \end{matrix}  (1)

1“*” This work has been partially performed as a part of a National Research Project supported by M.U.R.S.T. and supported by the PROGRAMMA DI SCAMBI INTER-NAZIONALI PER LA BREVE MOBILITA’ instituted by UNIVERSITA’ DI NAPOLI ”FEDERICO II”