Partial regularity for variational integrals with Morrey-H\"older zero-order terms, and the limit exponent in Massari's regularity theorem

TL;DR

This paper refines the partial regularity theory for variational integrals with zero-order terms, establishing sharp H"older exponents and confirming optimal regularity in the context of prescribed-mean-curvature hypersurfaces.

Contribution

It provides a detailed analysis of the dependence of regularity exponents on structural assumptions, extending Massari's theorem to optimal regularity limits.

Findings

Sharp H"older exponent dependence established

Partial regularity results extended to zero-order terms

Optimal regularity confirmed for prescribed-mean-curvature hypersurfaces

Abstract

We revisit the partial $\mathrm{C}^{1,\alpha}$ regularity theory for minimizers of non-parametric integrals with emphasis on sharp dependence of the H\"older exponent $\alpha$ on structural assumptions for general zero-order terms. A particular case of our conclusions carries over to the parametric setting of Massari's regularity theorem for prescribed-mean-curvature hypersurfaces and there confirms optimal regularity up to the limit exponent.