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1996 Local $T$-sets and renormalized solutions of degenerate quasilinear elliptic equations with an $L^1$-datum
Youcef Atik, Jean Michel Rakotoson
Adv. Differential Equations 1(6): 965-988 (1996).

Abstract

In this paper we study essentially the questions of uniqueness and stability of solutions of boundary value problems associated with equations of the type: $$ \text{div}(\hat a(x,u,\nabla u))+b(x)|u|^{\gamma-1}u=\mu\in L^1(\Omega) $$ on an arbitrary open subset $\Omega$ of $\mathbb{R}^N$ with $\hat a(x,u,\nabla u)$ a Carathéodory nonlinear function satisfying the general conditions of Leray-Lions where the coerciveness condition is weakened to allow degeneracies and becomes $$ \hat a(x,u,\xi)\cdot\xi\ge a(x)|\xi|^p,\ \forall u\in\mathbb{R},\ \forall\xi\in\mathbb{R}^N, \ \mbox{and $x$ a.e. in } \Omega, $$ with $p>1$ an arbitrary real number and $a$ an $L^1$-weight which might vanish or go to infinity on $S$, a closed subset of $\overline\Omega$ whose measure is zero. Here $b$ is an $L^1$-nonnegative function with properties similar to those of the weight $a$, $\gamma$ a positive number belonging to suitable intervals. For $p>1$ arbitrary and general weight $a$, we need new functional sets called "local T-sets" which are extensions of local Sobolev spaces. The "localization" is to handle the degeneracy. We get uniqueness and stability for $S$ satisfying a geometrical condition or $S$ and $a$ an analytic-geometrical one.

Citation

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Youcef Atik. Jean Michel Rakotoson. "Local $T$-sets and renormalized solutions of degenerate quasilinear elliptic equations with an $L^1$-datum." Adv. Differential Equations 1 (6) 965 - 988, 1996.

Information

Published: 1996
First available in Project Euclid: 25 April 2013

zbMATH: 0885.35040
MathSciNet: MR1409895

Subjects:
Primary: 35J60

Rights: Copyright © 1996 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.1 • No. 6 • 1996
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