Compensation phenomena for concentration effects via nonlinear elliptic estimates

Abstract

We study compensation phenomena for fields satisfying both a pointwise and a linear differential constraint. The compensation 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, focusing on the case where the cone is a subset of the space of symmetric matrices and the differential operator is 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 appends the classical compensated compactness framework for oscillations with a variant designed for concentrations, and also 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. k 3.2. A quantitative characterisation of null Lagrangians 3.3. k-Hessian equations 4. Elliptic estimates in two dimensions 4.1. Elliptic estimates for the divergence 4.2. Elliptic estimates for the curl 4.3. Optimality 4.4. Density of smooth functions and separation 5. New examples of div-quasiconcave integrands 6. Elliptic estimates for the divergence 6.1. Elliptic estimates 6.2. The case of exact constraints: optimality and higher integrability 7. Elliptic estimates for the curl 7.1. Elliptic estimates 7.2. The case of exact constraints: optimality and higher integrability 8. Other examples 8.1. A degenerate example 8.2. A second order example 8.3. A first order example Appendix A.