January/February 2012 On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to $1$ in a subdomain
Anna Mercaldo, Julio D. Rossi, Sergio Segura de León, Cristina Trombetti
Differential Integral Equations 25(1/2): 53-74 (January/February 2012). DOI: 10.57262/die/1356012825


In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain $\Omega$, which is the $p$--Laplacian equation, $-\mbox{div}(|\nabla u|^{p-2} \nabla u) =f$, in a subdomain $\Omega_1$ of $\Omega$ and the Laplace equation, $-\Delta u = f$, in its complementary (that is, our equation involves the so-called $p(x)$--Laplacian with a discontinuous exponent). We assume that the right-hand side $f$ belongs to $L^\infty(\Omega)$. For this problem, we study the behaviour of the solutions as $p$ goes to $1$, showing that they converge to a function $u$, which is almost everywhere finite when the size of the datum $f$ is small enough. Moreover, we prove that this $u$ is a solution of a limit problem involving the $1$-Laplacian operator in $\Omega_1$. We also discuss uniqueness under a favorable geometry.


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Anna Mercaldo. Julio D. Rossi. Sergio Segura de León. Cristina Trombetti. "On the behaviour of solutions to the Dirichlet problem for the $p(x)$-Laplacian when $p(x)$ goes to $1$ in a subdomain." Differential Integral Equations 25 (1/2) 53 - 74, January/February 2012. https://doi.org/10.57262/die/1356012825


Published: January/February 2012
First available in Project Euclid: 20 December 2012

zbMATH: 1249.35121
MathSciNet: MR2906546
Digital Object Identifier: 10.57262/die/1356012825

Primary: 35D30 , 35J66 , 35J75

Rights: Copyright © 2012 Khayyam Publishing, Inc.


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Vol.25 • No. 1/2 • January/February 2012
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