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Compensation phenomena for concentration effects via nonlinear elliptic estimates

André Guerra, Bogdan Raiţă, Matthew Schrecker

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.

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. 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.
Paper Structure (29 sections, 55 theorems, 307 equations, 1 figure, 1 table)

This paper contains 29 sections, 55 theorems, 307 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Nonlinear elliptic estimates
  3. The case of exact constraints: quasiconcavity and optimality
  4. Double cones and non-convex constraints
  5. Summary, ideas of proof, and open problems
  6. Outline of the paper
  7. Notation
  8. Symmetric cones and $k$-Hessian equations
  9. Symmetric polynomials and the cones $\Gamma_k$ and $\Gamma_k^*$
  10. A quantitative characterisation of null Lagrangians
  11. $k$-Hessian equations
  12. Elliptic estimates in two dimensions
  13. Elliptic estimates for the divergence
  14. Elliptic estimates for the curl
  15. Optimality
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\mathcal{K}\equiv \{A\in \varmathbb{R}^{2\times 2}: a_{11},a_{22}\geqslant 0\}$ and let $A_1,A_2$ be the rows of $A\in \varmathbb{R}^{2\times 2}$. Then In particular, writing $Q_2^+(K)=\{A\in \varmathbb{R}^{2\times 2}: \|A\|^2\leqslant K \det A\}$, we have

Figures (1)

  • Figure :

Theorems & Definitions (65)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Conjecture 1.1
  • Theorem 6
  • Proposition 7
  • Lemma 3.1: Characterization of $\Gamma_k$
  • Lemma 3.2
  • ...and 55 more