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

Umberto Guarnotta and Sunra Mosconi

Vol. 16 (2023), No. 8, 1955–1988
Abstract

For solutions of Div (DF(Du)) = f we show that the quasiconformality of zDF(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 two applications: a nonlinear Cordes condition for equations in divergence form and some partial results on the Cp conjecture.

Keywords
elliptic equations in divergence form, regularity, quasiconformal maps, Calderón–Zygmund estimates, Cordes condition
Mathematical Subject Classification
Primary: 30C65, 35B65, 35J62
Secondary: 49K20
Milestones
Received: 21 June 2021
Revised: 18 January 2022
Accepted: 7 March 2022
Published: 16 October 2023
Authors
Umberto Guarnotta
Dipartimento di Matematica e Informatica
Università degli Studi di Catania
Catania
Italy
Sunra Mosconi
Dipartimento di Matematica e Informatica
Università degli Studi di Catania
Catania
Italy

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