Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Wednesday, July 14, 16:00 ~ 16:30 UTC-3

On the Monge-Ampere system

Marta Lewicka

University of Pittsburgh, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Monge-Ampere equation $\det\nabla^2 u =f$ posed on a $N=2$ dimensional domain, has a natural weak formulation that appears as the constraint condition in the $\Gamma$-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: $curl^2 (\nabla v\otimes \nabla v) = -2f$ and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in $C^0(\omega)$, at the regularity $C^{1,\alpha}$ for any $\alpha<1/7$. $\\$

Does a similar result hold in higher dimensions $N>2$? Indeed it does, but one has to replace the Monge-Ampere equation by a {\em “Monge-Ampere system”}, altering $curl^2$ to the corresponding operator whose kernel consists of the symmetrised gradients of $N$-dimensional displacement fields. We will show how this Monge-Ampere system arises from the prescribed Riemannian curvature problem by matched asymptotic expansions, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d, and prove that its flexibility at $C^{1,\alpha}$ for any $\alpha<1/(N^2+N+1)$.

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