On nodal sets for Dirac and Laplace operators

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension. We also see that the nodal set of a $\Delta$-harmonic differential form on a closed manifold has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal.Comment: LaTeX, uses pstricks macro-package, 15 pages with 2 figures; to appear in Commun. Math. Phy

Some properties of solutions to weakly hypoelliptic equations

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which cover all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p-solution must vanish.Comment: published version (up to cosmetic issues

Zero Sets of Solutions to Semilinear Elliptic Systems of First Order

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an $n$-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth $(n-2)$-dimensional submanifolds. Hence it is countably $(n-2)$-rectifiable and its Hausdorff dimension is at most $n-2$. Moreover, it has locally finite $(n-2)$-dimensional Hausdorff measure. We show by example that every real number between 0 and $n-2$ actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order.Comment: 16 pages, LaTeX2e, 2 figs, uses pstricks macro packag