p(⋅)-Laplacian in ℝn2022/05/17 by Zachary Forrest, Forrest, Zachary, Robert D. Freeman +1
Computer Science · Mathematics · #22E25 (Secondary) #35D40 31C45 #35H20 (Primary) 31E05 #53C17 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2205.08612
openalex publication_date 2022/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the p(⋅)-Laplace equation in ℝn. The proof of the equivalence in the variable exponent case in Euclidean space was first given by Juutinen, Lukkari, and Parviainen (2010) and extended the equivalence of potential theoretic weak solutions and viscosity solutions to the p-Laplace equation in ℝn, given by Juutinen, Lindqvist, and Manfredi (2001). In both the fixed exponent case and the variable exponent case, the main argument is based on the maximum principle for semicontinuous functions, several approximations, and also applied the full uniqueness machinery of the theory of viscosity solutions. This paper extends the approach of Julin and Juutinen (2012) for the fixed exponent case, and so we employ infimal convolutions to present a direct, new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the p(⋅)-Laplace equation in ℝn.