# 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}
}
• 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… 11 Citations • Mathematics • 2022 . In this article we study functionals of the following type ˆ Ω (cid:16) h A ( x,u ) ∇ u, ∇ u i + Λ( x, u ) (cid:17) dx here A ( x,u ) = A + ( x ) χ { u> 0 } + A − ( x ) χ { u ≤ 0 } for some Abstract This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, where and are uniformly elliptic, Dini continuous coefficient matrices, • Mathematics • 2020 We study a free transmission problem driven by degenerate fully nonlinear operators. By framing the equation in the context of viscosity inequalities, we produce optimal regularity results for • Mathematics • 2020 We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogeneously, our analysis is two-fold: we study the regularity of • Mathematics Research in the Mathematical Sciences • 2021 It is shown that every penetrable obstacle with real-analytic boundary admits such an incident wave that does not scatter, and that there is a dichotomy for boundary points of any penetrables obstacle having this property. • Mathematics Journal of the European Mathematical Society • 2022 We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive • Mathematics • 2023 We study existence and regularity of weak solutions for the following PDE $$-\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in B_1}.$$ where$A(x,s) =
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In this paper, we consider a double-phase problem characterised by a transmission that takes place across the zero level"surface"of the minimiser of the functional  J(v,\Omega) = \int_\Omega \left(

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