Unique continuation for nonlinear variational problems

TL;DR

This paper investigates the unique continuation properties of solutions to nonlinear variational problems, demonstrating that weak and strong unique continuation hold for a broad class of nonlinear functionals using Almgren's frequency formula.

Contribution

It extends unique continuation results to nonlinear variational problems with complex growth conditions, including double phase and multiphase functionals.

Findings

Proves weak and strong unique continuation for nonlinear variational solutions.

Establishes estimates on the dimension of the vanishing set of solutions and their gradients.

Applies Almgren's frequency formula to nonlinear variational contexts.

Abstract

This paper is dedicated to the unique continuation properties of the solutions to nonlinear variational problems. Our analysis covers the case of nonlinear autonomous functionals depending on the gradient, as well as more general double phase and multiphase functionals with $(2,q)$-growth in the gradient. We show that all these cases fall in a class of nonlinear functionals for which we are able to prove weak and strong unique continuation via the almost-monotonicity of Almgren's frequency formula. As a consequence, we obtain estimates on the dimension of the set of points at which both the solution and its gradient vanish.