Viscosity solutions to the infinity Laplacian equation with strong absorptions

Abstract

<p style='text-indent:20px;'>In this paper, we study the Dirichlet problem of the inhomogeneous infinity Laplace equation with strong absorptions:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \ \ \Delta_{\infty}u(x)-\lambda(x)(u(x))^{\gamma}_+ = f(x), \ &{\rm{in}}\ \Omega, \\ \ \ u(x) = \phi(x), \ &{\rm{on}}\ \partial\Omega, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \gamma\in(0, 3). $\end{document}</tex-math></inline-formula> We first prove the existence and uniqueness of the viscosity solution to the Dirichlet problem. Then, under additional structure conditions on <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula>, we establish the <inline-formula><tex-math id="M3">\begin{document}$ C^{\frac{4}{3-\gamma}} $\end{document}</tex-math></inline-formula> regularity of the viscosity solution across the free boundary and the non-degeneracy. Based on the comparison principle, we also obtain the stability result, that is the viscosity solution of the Dirichlet problem converges uniformly to the solution to the corresponding Dirichlet problem of the homogeneous equation when the inhomogeneous term converges uniformly to <inline-formula><tex-math id="M4">\begin{document}$ 0. $\end{document}</tex-math></inline-formula></p>