Viscosity solutions to the infinity Laplacian equation with lower terms

Pub Date : 2023-06-25 DOI:10.58997/ejde.2023.42
Cuicui Li, Fang Liu
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引用次数: 1

Abstract

We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$\displaylines{ \Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr u=q, \quad\hbox{on }\partial\Omega,}$$ 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.
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无穷低项拉普拉斯方程的粘度解
我们建立了Dirichlet问题$$\displaylines{\Delta_\infty^h u=f(x,u),\quad\hbox{in}\Omega,\cr u=q,\quad \hbox{on}\partial \Omega}$$的粘性解的存在性和唯一性,其中\。算子\(\Delta_\infty u=\langle D^2uDu,Du\rangle\)是强退化、拟线性的无穷远拉普拉斯算子,它与绝对最小化Lipschitz扩张有关。当非齐次项\(f(x,t)\)在\(t)中不递减时,我们用Perron方法证明了粘性解的存在性。基于粘性解的摄动分析,我们还建立了一个唯一性结果。如果函数\(f(x,t)\)在\(t)中是非正(非负)且不递增的,则在域具有小直径的条件下,我们还通过迭代技术给出了粘性解的存在性。此外,我们还研究了当域满足某些正则条件时,具有奇异性$$\displaylines{\Delta_\infty^hu=-b(x)g(u),\quad\hbox{in}\Omega,\cr u>0,\quad \hbox{in}\Omega、\cr u=0,\quad \hbox{on}\ partial \ Omega,}$$边值问题粘性解的存在性和唯一性。我们分析了边界附近粘性解的渐近估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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