Classification of singular sets of solutions to elliptic equations

Abstract

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 $ \varepsilon $-regularity lemma for $ j $-symmetric homogeneous harmonic polynomial $ P $ with origin $ 0 $ as the isolated critical point in $ \mathbb{R}^{n-j} $, and by the generalized $ \varepsilon $-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.