Compensation phenomena for concentration effects via nonlinear elliptic estimates

TL;DR

This paper investigates nonlinear elliptic estimates that demonstrate how constraining fields to lie within certain cones can compensate for the lack of ellipticity, revealing new examples and challenging previous conjectures.

Contribution

It introduces new compensation phenomena for geometric cones and operators, extending the theory of compensated integrability and providing counterexamples to recent conjectures.

Findings

Maximal gain of integrability depends on both the operator and the cone

New examples of compensation phenomena for divergence and curl operators

Identification of Div-quasiconcave integrands under convex constraints

Abstract

We study compensation phenomena for fields satisfying both a pointwise and a linear differential constraint. This effect takes the form of nonlinear elliptic estimates, where constraining the values of the field to lie in a cone compensates for the lack of ellipticity of the differential operator. We give a series of new examples of this phenomenon for a geometric class of cones and operators such as the divergence or the curl. One of our main findings is that the maximal gain of integrability is tied to both the differential operator and the cone, contradicting in particular a recent conjecture from arXiv:2106.03077. This extends the recent theory of compensated integrability due to D. Serre. In particular, we find a new family of integrands that are Div-quasiconcave under convex constraints.