arXiv subjects

Jesus Garcia Azorero

Publications and source records attributed to Jesus Garcia Azorero.

2 recordsLinked to original sources

arXiv2014

An existence result for the infinity laplacian with non-homogeneous Neumann boundary conditions using Tug-of-War games

In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity solution of the mixed boundary value problem, $$ \{\begin{array}{ll} \displaystyle -Δ_{\infty}u(x)=0\quad & \text{in} Ω, \displaystyle \frac{\partial u}{\partial n}(x)= g (x)\quad & \text{on} Γ_N, \displaystyle u(x)= 0 \quad & \text{on} Γ_D. \end{array}. $$

math.AP↗

arXiv2009

A mixed problem for the infinity laplacian via Tug-of-War games

In this paper we prove that a function $ u\in\mathcal{C}(\barΩ)$ is the continuous value of the Tug-of-War game described in \cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-Δ_{\infty}u(x)=0\quad & \text{in} Ω, \frac{\partial u}{\partial n}(x)=0\quad & \text{on} Γ_N, u(x)=F(x)\quad & \text{on} Γ_D. By using the results in \cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\it absolutely minimizing Lipschitz extension} to the whole $\barΩ$ (in the sense of \cite{Aronsson} and \cite{PSSW}) of the boundary data $ F:Γ_D\to\R $.

math.AP↗