Nodal Sets for Broken Quasilinear Partial Differential Equations with Dini Coefficients

TL;DR

This paper investigates the structure and measure of nodal sets for solutions to a class of broken quasilinear PDEs with Dini continuous coefficients, introducing new approximation techniques and regularity results.

Contribution

It develops a novel iteration method for higher-order approximation at singular points and establishes a structure theorem for the singular set of solutions.

Findings

Established a structure theorem for singular sets.

Estimated Hausdorff measure of nodal sets under bounded vanishing order.

Proved Lipschitz regularity and differentiability of solutions and nodal sets.

Abstract

This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, \begin{equation*} \mbox{div} (a_+ \nabla u^+ - a_- \nabla u^-) = \mbox{div} f, \end{equation*} where $a_+$ and $a_-$ are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that $a_+$ and $a_-$ are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below H\"older regime, and as a result, we establish a structure theorem for singular sets. We also estimate the Hausdorff measure of nodal sets, provided that the vanishing order of given solution is bounded throughout its nodal set, via an approach that extends the classical argument to certain solutions with discontinuous gradient. Besides, we also prove Lipschitz regularity of solutions and continuous differentiability of their nodal set around regular points.