Partial Regularity Results for Quasilinear Systems

Abstract

We next study more general quasilinear systems in divergence form and deal with partial $C^{1,\alpha }$-regularity results for their weak solutions. More precisely, we first give a basic higher integrability statement. Then we employ the method of $\mathcal{A}$-harmonic approximation, which was introduced in the previous chapter, in order to prove in the first place the partial C ¹-regularity of weak solutions outside of a singular set which is of $\mathcal{L}^{n}$-measure zero and in the second place the optimal regularity improvement from C ¹ to $C^{1,\alpha }$ for some α > 0 (determined by the regularity of the governing vector field). These results come along with a characterization of the exceptional set on which singularities of a weak solution may arise. However, it does not directly allow for a non-trivial bound on its Hausdorff dimensions, but this requires further work. In different settings, from simple to quite general ones, we explain (fractional) higher differentiability estimates for the gradient of weak solutions. These provide, in turn, the desired bounds for the Hausdorff dimension of the singular set.