Foundations for the Vectorial Case

Abstract

In this chapter we continue to investigate the regularity for weak solutions, but now we address the case of vector-valued solutions where we encounter fundamentally new phenomena when compared to the scalar case. In order to concentrate on the central concepts and ideas, we here restrict ourselves to the model case of quasilinear systems that are linear in the gradient variable. We first give two examples of elliptic systems, which admit a discontinuous or even unbounded weak solution. Then we investigate the optimal regularity of weak solutions in dependency of the “degree” of nonlinearity of the governing vector field. In this regard, we start by discussing the linear theory and establish full regularity estimates. This is quite peculiar and a consequence of the particular structure of the coefficients, since for more general systems, as in the counterexamples, one merely expects partial regularity results, that is, regularity outside of negligible sets. Secondly, we present three different strategies for proving partial $C^{0,\alpha }$-regularity results for such systems, where the coefficients may depend also explicitly on the weak solution. More precisely, we explain the main ideas for the blow-up technique, the method of $\mathcal{A}$-harmonic approximation, and the indirect approach.