Existence results for a fourth order partial differential equation arising in condensed matter physics

TL;DR

This paper investigates a fourth order semilinear PDE from condensed matter physics, analyzing stationary and parabolic solutions in various domains, and discusses open problems on self-similar solutions.

Contribution

It provides new existence results for solutions of a complex fourth order PDE in arbitrary and symmetric domains, and explores the initial-boundary value problem.

Findings

Existence of stationary solutions in arbitrary domains.

Existence of radially symmetric solutions in the unit ball.

Open problem on self-similar solutions.

Abstract

We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.