H\"older continuity of bounded, weak solutions of a variational system in the critical case

TL;DR

This paper proves that bounded, weak solutions to a certain class of variational systems in two dimensions are H"older continuous, even without smallness restrictions on inhomogeneity, addressing a critical case in regularity theory.

Contribution

It establishes H"older continuity for bounded weak solutions of variational systems in the critical 2D setting without smallness assumptions on inhomogeneity.

Findings

Bounded weak solutions are H"older continuous.

Results apply in the critical dimension without smallness conditions.

Addresses an open problem in regularity of variational systems.

Abstract

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(\Omega;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x, u, Du)\quad\text{in }\Omega. Under natural growth conditions on the principal part and the inhomogeneity, but without any further restriction on the growth of the inhomogeneity (for example, via a smallness condition), we use a blow-up argument to prove that every bounded, weak solution of the system is H\"older continuous. Since the dimension of $\Omega$ is $2$ and $u\in W^{1, 2}(\Omega;\mathbb{R}^N)$, we are in the critical setting, and hence, cannot use the Sobolev embedding theorem to deduce H\"older continuity. Our results are connected to a particular case of the open problem of whether all solutions (and not just extremals) of variational systems are H\"older continuous in the critical setting.