The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

TL;DR

This paper investigates the existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation in critical and supercritical Sobolev spaces, revealing generic non-existence results.

Contribution

It demonstrates that for a generic set of functions in certain Lebesgue and Orlicz spaces, solutions with expected regularity do not exist, advancing understanding of the Jacobian equation's behavior.

Findings

No solutions with expected regularity for generic functions in L^p and L log L spaces.

Existence and regularity results depend on the function space and genericity.

Highlights limitations of regularity theory for the prescribed Jacobian equation.

Abstract

We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $\det Du = f$, where $f$ is integrable and bounded away from zero. In particular, we take $f\in L^p$, where $p > 1$, or in $L\log L$. We prove that for a Baire-generic $f$ in either space there are no solutions with the expected regularity.