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

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.