• DOI:10.1512/iumj.2019.68.7711
• Corpus ID: 119595558

Nodal sets for "broken" quasilinear pdes

@article{Kim2017NodalSF,
  title={Nodal sets for "broken" quasilinear pdes},
  author={Sunghan Kim and Ki-ahm Lee and Henrik Shahgholian},
  journal={Indiana University Mathematics Journal},
  year={2017}
}

• Sunghan Kim, Ki-ahm Lee, H. Shahgholian
• Published 31 May 2017
• Mathematics
• Indiana University Mathematics Journal

We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, \begin{equation*} \operatorname{div}(A_s(x,u)\nabla u)=\operatorname{div}{\vec f}(x), \end{equation*} where $A_s(x,u)$ has "broken" derivatives of order $s\geq 0$, such as \begin{equation*} A_s(x,u) = a(x) + b(x)(u^+)^s, \end{equation*} with $(u^+)^0$ being understood as the characteristic function on $\{u>0\}$. The vector ${\vec f}(x)$ is assumed to be $C^\alpha$ in case $s=0$, and $C^{1,\alpha}$ (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.