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

Abstract: In this paper, we study flexibility of the weak solutions to the Monge-Amp`ere system (MA) via convex integration. This 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 regular $\mathcal{C}^{1,\alpha}$ solutions to the weak formulation (VK) of (MA), for all $\alpha<\frac{1}{1+d(d+1)/k}$ where $d$ is an arbitrary dimension and $k$ is an arbitrary codimension of the problem. This 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 $\alpha <1$ proved by K\"allen, strictly requires a large codimension $k\geq d(d+1)$. We also reproduce K\"allen's result in the context of (MA). At $k=1$, our result 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. Finally, our results extend the initial findings by the author and Pakzad for (MA) and $d=2, k=1$. 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.