Equivalence of viscosity and weak solutions for the $p\left(x\right)$-Laplacian

Juutinen, Petri; Lukkari, Teemu; Parviainen, Mikko

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1471-1487

We consider different notions of solutions to the $p\left(x\right)$-Laplace equation

$$-\mathrm{div}\left(\right|Du\left(x\right){|}^{p\left(x\right)-2}Du\left(x\right))=0$$

with $1<p\left(x\right)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p\left(x\right)$-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 Radó type removability theorem.

Classification : 35J92, 35D40, 31C45, 35B60
Keywords: Comparison principle, Viscosity solutions, Uniqueness, $ p(x)$-Superharmonic functions, Radó type theorem, Removability

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