$\mathcal{A}$-harmonic approximation and partial regularity, revisited

TL;DR

This paper presents a new, elementary approach to establishing partial regularity of local minima in quasiconvex variational problems, avoiding complex tools used in prior methods.

Contribution

It introduces a direct harmonic approximation lemma that simplifies the proof of partial regularity for quasiconvex integrals, relying solely on basic elliptic theory and Sobolev embeddings.

Findings

Proves partial regularity results for local minima of quasiconvex integrals.

Develops a direct harmonic approximation lemma without Lipschitz truncations.

Simplifies the methodology for regularity in calculus of variations.

Abstract

We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,\alpha}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.