Submanifolds that are level sets of solutions to a second-order elliptic PDE

TL;DR

This paper characterizes which noncompact hypersurfaces can be transformed into level sets of harmonic functions using diffeomorphisms, providing new methods for constructing solutions and analyzing intersections of level sets.

Contribution

It introduces a versatile sufficient condition showing algebraic noncompact hypersurfaces can be transformed into harmonic function level sets via diffeomorphisms, advancing the understanding of PDE level sets.

Findings

Any algebraic noncompact hypersurface can be transformed into a harmonic function level set.

Developed a new technique combining local constructions with global approximation theorems.

Studied intersections of level sets with applications to a problem of Berry and Dennis.

Abstract

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any (possibly disconnected) algebraic noncompact hypersurface can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of $\RR^n$. The technique we use, which is a significant improvement of the basic strategy we recently applied to construct solutions to the Euler equation with knotted stream lines (Ann. of Math., in press), combines robust but not explicit local constructions with appropriate global approximation theorems. In view of applications to a problem of Berry and Dennis, intersections of level sets are also studied.