Equivalence and regularity of weak and viscosity solutions for the anisotropic $${{\textbf {p}}}(\cdot )$$ -Laplacian

Abstract

In this paper, we state the equivalence between weak and viscosity solutions for non-homogeneous problems involving the anisotropic \({{\textbf {p}}}(\cdot )\)-Laplacian. The proof that viscosity solutions are weak solutions is performed by the inf-convolution technique. However, due to the anisotropic nature of the \({{\textbf {p}}}(\cdot )\)-Laplacian we adapt the definition of inf-convolution to the non-homogeneity of this operator. For the converse, we develop comparison principles for weak solutions. Since the locally Lipschitz assumption is crucial to get the viscosity-weak implication, we prove that a class of bounded viscosity solutions are indeed locally Lipschitz.