Nodal sets for "broken" quasilinear pdes

Abstract

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part,is assumed to be C α in case s = 0, and C 1,α (or higher) in case s > 0.Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

Nodal sets for broken quasilinear partial differential equations with Dini coefficients

“…In addition, we also prove the continuous differentiability of the nodal set around regular points. In particular, the modulus of continuity of the normal mapping becomes H€ older when both coefficients a þ and a À are assumed to be H€ older, hence generalising the corresponding result of [4]. More specifically, we show the following.…”

“…In this paper, we extend the main results in [4] from H€ older to the Dini regime. More exactly, our result concerning the Lipschitz regularity of solutions is stated as follows:…”

“…Our analysis on the singular set resembles the author's earlier collaboration [4] with Lee and Shahgholian, in the sense that here we also establish uniform point-wise approximation (Lemma 3.3 and Lemma 5.8) for the corrected version, u þ À jðzÞu À , for each point z. However, we develop a new iteration argument (Proposition 3.1 and…”

“…Proposition 5.6) to perform the uniform approximation, since our coefficients a þ and a À are assumed to be Dini continuous, while the argument involved in [4] is only available for H€ older coefficients.…”

“…In a more recent collaboration [4], the authors observed that after a change of coordinates, function v z ¼ u þ À jðzÞu À , for each interior point z, satisfies…”