## View abstract

### 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. document.getElementById('cloakdd6b8d0eb7aed040bc4b9811b515c14a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addydd6b8d0eb7aed040bc4b9811b515c14a = 'l&#101;w&#105;ck&#97;' + '&#64;'; addydd6b8d0eb7aed040bc4b9811b515c14a = addydd6b8d0eb7aed040bc4b9811b515c14a + 'p&#105;tt' + '&#46;' + '&#101;d&#117;'; var addy_textdd6b8d0eb7aed040bc4b9811b515c14a = 'l&#101;w&#105;ck&#97;' + '&#64;' + 'p&#105;tt' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakdd6b8d0eb7aed040bc4b9811b515c14a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addydd6b8d0eb7aed040bc4b9811b515c14a + '\'>'+addy_textdd6b8d0eb7aed040bc4b9811b515c14a+'<\/a>';

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 $\alpha<1/(N^2+N+1)$.