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

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

2010-07-29

arxiv.org/abs/1007.5181v2

Mathematics

Analysis of PDEs

46 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Submanifolds that are level sets of solutions to a second-order elliptic PDE does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Submanifolds that are level sets of solutions to a second-order elliptic PDE, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Submanifolds that are level sets of solutions to a second-order elliptic PDE will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-451170