Exact solution for the p-Laplace minimization on an L-shaped domain

Determine an explicit analytical expression for the unique minimizer u of the p-Laplace variational problem J(v) = (1/p) ∫_Ω ||∇v||^p dx − ∫_Ω f v dx with p = 3 on the L-shaped planar domain Ω = [0,2]^2 minus [1,2]^2, subject to homogeneous Dirichlet boundary conditions and constant load f(x) = −10; that is, find the exact solution u in V = W^{1,p}_0(Ω) for this configuration.

Background

The paper studies three benchmark nonlinear energy minimization problems and compares MATLAB and Python implementations. For the p-Laplace case, the authors consider Ω = [0,2]^2 \ [1,2]^2, p = 3, and a constant right-hand side f = −10 with homogeneous Dirichlet boundary conditions.

While the finite element solution is used as a numerical benchmark, the authors explicitly note that the exact (analytical) solution u for this specific setting is unknown, despite the general theoretical fact that the minimizer is unique for p > 1. This makes the determination of a closed-form or exact expression for u an unresolved question in the context of this benchmark.