Nodal sets of solutions of elliptic and parabolic equations

Some geometrical properties such as convexity and star-shapedness of level sets of positive solutions to elliptic or parabolic equations have been studied via various authors (see an excellent lecture set of notes by B. Kawohl, [ 81 and the references therein). One reason to choose star-shapedness and convexity, other than other properties, as geometrical properties, is that they can be easily described and are accessible to variational and maximum principles in analysis. Here we are interested in some general questions regarding level sets of solutions to elliptic and parabolic equations, such as the size (i.e., HausdorB measure of appropriate dimension) and the topology of these level sets and estimates on critical point sets, etc. The study of such problems was motivated by the study of moving defects in evolution problems of harmonic maps and liquid crystals (see [lo]). In [lo], I have studied a model for the evolution of nematic liquid crystals. The singular set of optical axes (i.e., defects) of liquid crystal in motion can be described precisely by the nodal set of solutions to certain parabolic equations. Recently, there were several rather interesting articles studying the nodal sets of eigenfunctions of Laplacians on a compact Riemannian manifold by Donnelly and Feffermann [ 2 1, [ 3 ] or, generally, solutions of second-order elliptic equations in Hardt and Simon [ 61. The present work can also be viewed as a natural extension of [2], [3], and [ 61. Our main result can be stated as: