Entropy solutions for the 𝑝(𝑥)-Laplace equation

Sanchón, Manel; Urbano, José Miguel

doi:10.1090/S0002-9947-09-04399-2

Entropy solutions for the $p(x)$-Laplace equation
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by Manel Sanchón and José Miguel Urbano PDF

Trans. Amer. Math. Soc. 361 (2009), 6387-6405 Request permission

Abstract:

We consider a Dirichlet problem in divergence form with variable growth, modeled on the $p(x)$-Laplace equation. We obtain existence and uniqueness of an entropy solution for $L^1$ data, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponent, for which we obtain new inclusion results of independent interest.

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