  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.
