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.
| Original language | English |
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| Article number | 1 |
| Number of pages | 56 |
| Journal | Ars Inveniendi Analytica |
| Early online date | 22 Feb 2024 |
| DOIs | |
| Publication status | Published - 22 Feb 2024 |
Funding
A. Guerra was supported by the Infosys Membership at the Institute for Advanced Study. M. Schrecker’s research is supported by the EPSRC Early Career Fellowship EP/S02218X/1. We thank Luc Nguyen for helpful discussions concerning k-Hessian equations and Denis Serre for his interesting comments. We also thank the anonymous referees for the careful reading of the manuscript and comments, which greatly improved the quality of the paper.