Abstract

This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, $$\text{div}(a_{+}\nabla u^{+}-a_{-}\nabla u^{-})=\text{div}\mathrm{ }f,$$ 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 Hölder 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.

摘要

本文关注的是一个破拟线性偏微分方程的弱解的节点集， $$\text{div}({\mathrm{一种}}_{+}\nabla {你}^{+}-{\mathrm{一种}}_{-}\nabla {你}^{-})=\text{div}\mathrm{ }F,$$ 在哪里 ${\mathrm{一种}}_{+}$ 和 ${\mathrm{一种}}_{-}$ 是一致椭圆的 Dini 连续系数矩阵，受强相关性的约束，即 ${\mathrm{一种}}_{+}$ 和 ${\mathrm{一种}}_{-}$是一些标量函数的倍数。在这样的结构条件下，我们开发了一个迭代论证来实现奇异点处解的高阶逼近，这对于 Hölder 制度以下的标准椭圆偏微分方程也是新的，因此，我们建立了奇异集的结构定理。我们还估计了节点集的 Hausdorff 测度，前提是给定解的消失阶在其整个节点集内是有界的，通过一种将经典论证扩展到具有不连续梯度的某些解的方法。此外，我们还证明了解的 Lipschitz 正则性和它们围绕正则点的节点集的连续可微性。