The Monge-Ampere system: convex integration in arbitrary dimension and codimension

Abstract

In this paper, we study flexibility of weak solutions to the Monge-Amp\`ere system (MA) via convex integration. This new system of Pdes is an extension of the Monge-Amp\`ere equation in $d=2$ dimensions, naturally arising from the prescribed curvature problem and closely related to the classical problem of isometric immersions (II). Our main result achieves density in the set of subsolutions, of the H\"older $\mathcal{C}^{1,\alpha}$ solutions to the Von K\'arm\'an system (VK) which is the weak formulation of (MA). The regularity exponent $\alpha$ is any exponent satisfying $\alpha<\frac{1}{1+ d(d+1)/k}$ where $d$ is an arbitrary dimension and $k$ an arbitrary codimension of the problem. At $k=1$, this agrees with the regularity $\mathcal{C}^{1,\alpha}$ for (II) with any $\alpha <\frac{1}{1+d(d+1)}$, proved by Conti, Delellis and Szekelyhidi. At $d=2, k=1$, this extends the initial findings by the author and Pakzad for (MA). Our result seems to be optimal, from the technical viewpoint, for the corrugation-based convex integration scheme. In particular, it covers the codimension interval $k\in \big(1, d(d+1)\big)$ so far uncharted even for the system (II), since the regularity $\mathcal{C}^{1,\alpha}$ with any $\alpha <1$ achieved by K\"allen in \cite{Kallen}, strictly requires a large codimension. Our second main result reproduces K\"allen's result in the context of (MA), obtaining density in the set of subsolutions, of $\mathcal{C}^{1,\alpha}$ regular solutions for any $\alpha<1$ whenever $k\geq d(d+1)$. As an application of our results for (VK), we derive an energy scaling bound in the quantitative immersability of Riemannian metrics, for nonlinear energy functionals modelled on the energies of deformations of thin prestrained films in the nonlinear elasticity.