Viscosity solutions to the infinity Laplacian equation with lower terms

Electronic Journal of Differential Equations

article

Cuicui Li¹  Fang Liu¹ 

[1] Department of Mathematics School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing 210094

关键词: Infinity Laplacian;    existence;    uniqueness;    asymptotic estimate;    viscosity solution.;   

DOI  :  10.58997/ejde.2023.42

学科分类：数学（综合）

来源: Texas State University

【 摘 要 】

We establish the existence and uniqueness of viscosity solutions to the Dirichlet problem $$\displaylines{\Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cru=q, \quad\hbox{on }\partial\Omega, }$$ where \(q\in C(\partial\Omega)\), \(h>1\), \(\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u\). The operator \(\Delta_\infty u=\langle D^2uDu,Du \rangle\) is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term \(f(x,t)\) is non-decreasing in \(t\), we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function \(f(x,t)\) is nonpositive (nonnegative) and non-increasing in \(t\), we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$\displaylines{\Delta_\infty^h u=-b(x)g(u), \quad \hbox{in } \Omega, \cr u>0, \quad \hbox{in } \Omega, \cr u=0, \quad \hbox{on }\partial\Omega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.

【 授权许可】

CC BY