论文标题
与DINI系数的折断的准部分偏微分方程的节点集
Nodal Sets for Broken Quasilinear Partial Differential Equations with Dini Coefficients
论文作者
论文摘要
本文与损坏的准部分部分微分方程的结节相关的解决方案有关，\ begin {equication*} \ mbox {div}（a_+ \ nabla u^+ - a_- a _- \ a_- \ nabla u^ - ） DINI连续系数矩阵，存在$ a _+$和$ a _- $的强相关性，这是彼此之间的某些标量功能的倍数。在这种结构条件下，我们开发了一个迭代论点，以实现奇异点上溶液的高阶近似，这对于低于Hölder制度的标准椭圆PDE也是新的，因此，我们为单数集建立了一个结构定理。我们还通过一种方法通过将经典参数扩展到具有不连续梯度的某些解决方案的方法来估算淋巴结集的Hausdorff度量。此外，我们还证明了Lipschitz规律性的解决方案以及它们的淋巴结的持续可不同性。
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 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.
