A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

TL;DR

This paper establishes that the quasiconformality of the derivative of the nonlinearity function is crucial for the Sobolev regularity of the stress field in divergence form elliptic equations, generalizing uniform ellipticity.

Contribution

It introduces a broad notion of uniform ellipticity based on quasiconformality, unifying known cases and providing optimality examples, with applications to operator locality, nonlinear Cordes conditions, and regularity conjectures.

Findings

Quasiconformality of DF(z) implies Sobolev regularity of DF(Du)

The introduced class generalizes all known uniform ellipticity cases

Applications include operator locality, nonlinear Cordes condition, and partial C^{p'}-conjecture results

Abstract

For solutions of ${\rm div}\,(DF(Du))=f$ we show that the quasiconformality of $z\mapsto DF(z)$ is the key property leading to the Sobolev regularity of the stress field $DF(Du)$, in relation with the summability of $f$. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present three applications: the study of the strong locality of the operator ${\rm div}\,(DF(Du))$, a nonlinear Cordes condition for equations in divergence form, and some partial results on the $C^{p'}$-conjecture.