Nodal lines of broken PDEs

Problem's Description

For $s$ a real number consider \begin{equation} \hbox{div}(A_s(x,u)\nabla u)= \hbox{div} ({\mathbf f}) \quad\text{in }B_1, \end{equation} where \begin{equation} A_s(x,u)= a(x) + b(x)(u^+)^s. \end{equation}

Here $a$ and $b$ satisfy \begin{equation} \lambda\leq a(x), b(x) \leq\frac{1}{\lambda}\quad (x\in\bar{B}_1), \end{equation} with a constant $0< \lambda< 1$.

This problem for the case $s>0$ was studied in [1]. For the case $s< 0$, and non-cconstant $a,b$ the problem seems quite challenging. To simplify the matter, let $a$ and $b$ be positive constants. Then $\phi(u;s)$, defined by \begin{equation*} \phi(u;s) = \begin{cases} au + \frac{b(u^+)^{s+1}}{s+1},&\text{if }s< 0\text{ and }s\neq -1,\\ au + b\log u^+,&\text{if }s = -1, \end{cases} \end{equation*} becomes harmonic in $B_1$. When $-1< s< 0$, one may deduce from the inverse function theorem that $u\in C^{1+ s}(B_1)$. However, when $s\leq -1$, the positive part blows up as $u$ approaches $0+$, and a set of singularities for $\phi(u;s)$ along $\partial \{u>0 \}$ arise.

Open problem: Existence, optimal regularity of the solutions, and that of the zero level surface of $u$.

1. ArticleIs an originNodal Sets for "Broken" Quasilinear PDEs

   • Sunghan Kim,
   • Ki-Ahm Lee,
   • Henrik Shahgholian

   arXiv

2. ArticleIs an originAn Elliptic Free Boundary Arising From the Jump of Conductivity

   • Sunghan Kim,
   • kiahm Lee,
   • Henrik Shahgholian

   arXiv