Nonsmooth Convex Functionals and Feeble Viscosity Solutions of singular Euler-Lagrange Equations

TL;DR

This paper establishes that continuous local minimisers of certain convex functionals are very weak viscosity solutions of singular Euler-Lagrange equations, introducing a novel regularisation method to handle strong singularities and extending classical existence results.

Contribution

It introduces systematic flat sup-convolution regularisations for singular PDEs and extends existence theorems in the calculus of variations for these equations.

Findings

Local minimisers are very weak viscosity solutions.

Regularisation method cancels strong singularities.

Extension of classical Dirichlet problem existence results.

Abstract

Let F be nonnegative, convex and smooth off a compact set K. We prove that continuous local minimisers of convex functionals are "very weak" viscosity solutions in the sense of Juutinen-Lindqvist of the highly singular Euler-Lagrange PDE expanded. The hypotheses on F do not guarrantee existence of minimising weak solutions and include the singular p-Laplacian for 1<p<2. A much deeper converse is also true, if K={0} and extra natural assumptions are satisfied. Our main advance is that we introduce systematic "flat" sup-convolution regularisations which apply to general singular nonlinear PDEs in order to cancel the strong singularity of F. As an application we extend a classical theorem of Calculus of Variations regarding existence for the Dirichlet problem. These results extends previous work of Julin-Juutinen and Juutinen-Lindqvist-Manfredi.