1-Laplacian Type Equations with Neumann Boundary Condition

Amir Moradifam

Published 2016-12-26Version 1

We study existence of solutions of the general 1-Laplacian type equation \begin{eqnarray}\label{abs} \left\{ \begin{array}{ll} \nabla_x \cdot \nabla_p \varphi(x,\frac{Du}{|Du|})=0 &\text{in } \Omega \left[ \nabla_p \varphi (x, \frac{Du}{|Du|}), \nu_{\Omega} \right]=\lambda g&\text{on }\partial \Omega, \end{array} \right. \end{eqnarray} where $\varphi(x,p)$ is convex, continuous, and homogeneous function of degree $1$ with respect to the $p$ variable, and $g$ satisfies the comparability condition $\int_{\partial \Omega} g dS=0$. We prove that for every $0\not \equiv g \in L^{\infty}(\partial \Omega)$ there exists a unique $\lambda^*>0$ for which the equation (\ref{abs}) has infinitely many solutions in $BV(\Omega)$, and if $\lambda \neq \lambda^*$, then (\ref{abs}) does not admit any solutions. In addition, when $\lambda=\lambda^*$, it is shown that there exists a divergence free vector field $T\in (L^{\infty}(\Omega))^n$ that determines the structure of level sets of all solutions of (\ref{abs}), i.e. $T$ determines $\frac{Du}{|Du|}$, $|Du|-$ a.e. in $\Omega$, for all solutions $u$. We also present a numerical algorithm which simultaneously finds $\lambda^*$, $T$, and a solution of the degenerate equation (\ref{abs}).