Lipschitz regularity for inner-variational equations

TL;DR

This paper establishes Lipschitz regularity for a broad class of nonlinear first-order PDEs derived from energy variational principles, highlighting the optimal regularity achievable even in simple models.

Contribution

It proves Lipschitz continuity for solutions of inner-variational equations, using topological methods, extending regularity results in nonlinear PDEs related to energy minimization.

Findings

Lipschitz regularity holds for solutions of inner-variational equations

Even in simple Dirichlet energy models, solutions are not necessarily differentiable everywhere

Topological arguments are essential in the proofs

Abstract

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to relay on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity; specifically, neo-Hookian type problems.