Singular Sets of Higher Order Elliptic Equations

The implicit function theorem implies that the zero set of a smooth function, the set where the function vanishes, is a smooth hypersurface away from the critical zero set. Hence to study zero sets it is important to understand the structure of the critical zero sets. For solutions of the second order elliptic equations the critical zero sets represent the singular parts of zero sets. They have the Hausdor dimension not greater than n 2. Hence sometimes they are called the singular sets of solutions. This result is not true for higher order elliptic equations. For example the critical zero sets may occupy the whole zero sets for some biharmonic polynomials. In order to study the zero sets of solutions of higher order elliptic equations we need to identify their singular parts and study their structure. The singular sets of solutions to elliptic equations of the second order have been studied by many people. In [19], Hardt and Simon proved that for classical solutions with relatively high order derivatives singular sets are countable unions of subsets of correspondingly smooth (n 2)-dimensional submanifolds. Thus, they are countably (n 2)-recti able. See also [5]. This result was generalized to weak solutions in [14]. It is proved that for weak solutions, as long as they do not vanish to in nite order, the singular sets are countable unions of subsets of C (n 2)-dimensional submanifolds, for some 2 (0; 1). Hence they are also countably (n 2)-recti able. Concerning the size of singular sets, M. and T. Ho mann-Ostenhof and N. Nadirashvili [20] showed that the singular sets of smooth solutions in three dimensional space have locally nite one dimensional Hausdor measure. It was generalized to arbitrary dimensions, independently by one of the authors in [18] and M. and T. Ho mann-Ostenhof and N. Nadirashvili