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arXiv · 1006.3482

Equivalence of Viscosity and Weak Solutions for the $p(x)$-Laplacian

Petri Juutinen·Teemu Lukkari·Mikko Parviainen
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Abstract

We consider different notions of solutions to the $p(x)$-Laplace equation $-\div(\abs{Du(x)}^{p(x)-2}Du(x))=0$ with $ 1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem.

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Petri Juutinen, Teemu Lukkari, Mikko Parviainen. 2010-06-17. Equivalence of Viscosity and Weak Solutions for the $p(x)$-Laplacian. https://doi.org/10.1016/j.anihpc.2010.09.004

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