Advances in Differential Equations

Inhomogeneous Dirichlet problems involving the infinity-Laplacian

Bhattacharya Bhattacharya and Ahmed Mohammed

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Abstract

Our purpose in this paper is to provide a self-contained account of the inhomogeneous Dirichlet problem ${\Delta_\infty} u=f(x,u)$ where $u$ represents prescribed continuous data on the boundary of bounded domains. We employ a combination of Perron's method and a priori estimates to give general sufficient conditions on the right-hand side $f$ that would ensure existence of viscosity solutions to the Dirichlet problem. Examples show that these sufficient conditions may not be relaxed. We also identify a class of inhomogeneous terms for which the corresponding Dirichlet problem has no solution in any domain with large in-radius. Several results, which are of independent interest, are developed to build towards the main results. The existence theorems provide a substantial improvement of previous results, including our earlier results [7] on this topic.

Article information

Source
Adv. Differential Equations Volume 17, Number 3/4 (2012), 225-266.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703086

Mathematical Reviews number (MathSciNet)
MR2919102

Zentralblatt MATH identifier
1258.35094

Subjects
Primary: 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations

Citation

Bhattacharya, Bhattacharya; Mohammed, Ahmed. Inhomogeneous Dirichlet problems involving the infinity-Laplacian. Adv. Differential Equations 17 (2012), no. 3/4, 225--266. https://projecteuclid.org/euclid.ade/1355703086.


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