CLASSIFICATION OF SINGULAR SETS OF SOLUTIONS TO ELLIPTIC EQUATIONS

In this paper, we mainly investigate the classification of singular sets of solutions to elliptic equations. Firstly, we define the j-symmetric singular set S-j (u) of solution u, and show that the Hausdorff dimension of the j-symmetric singular set S-j (u) is not more than j. Then we prove the generalized epsilon-regularity lemma for j-symmetric homogeneous harmonic polynomial P with origin 0 as the isolated critical point in Rn - j, and by the generalized epsilon-regularity lemma, we show the Hausdorff measure estimate of the j-symmetric singular set S-j(u). Moreover, we study the geometric structure of interior singular points of solutions u in a planar bounded domain.