On the existence of integrable solutions to nonlinear elliptic systems and variational problems with linear growth

TL;DR

This paper studies nonlinear elliptic systems with linear growth, showing that solutions with measure-valued properties can be understood as weak solutions under certain structural conditions, with boundary measures accounting for boundary conditions.

Contribution

It establishes conditions under which measure-valued solutions to nonlinear elliptic systems are equivalent to weak solutions, extending understanding of boundary measure effects.

Findings

Solutions can be interpreted as weak solutions under asymptotic Uhlenbeck structure.

Boundary conditions are represented by measures supported on boundary subsets.

The approach generalizes minimal surface equations to broader elliptic systems.

Abstract

We investigate the properties of certain elliptic systems leading, a~priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic Uhlenbeck structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously as in the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed.