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arXiv · 1104.1462

Inhomogeneous Dirichlet problems involving the infinity-Laplacian

Tilak Bhattacharya·Ahmed Mohammed
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Abstract

Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $Δ_\infty u=f(x,u)$ where $u$ takes a 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 substantial improvement of previous results, including our earlier results \cite{BMO} on this topic.

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Tilak Bhattacharya, Ahmed Mohammed. 2011-06-28. Inhomogeneous Dirichlet problems involving the infinity-Laplacian. https://arxiv.org/abs/1104.1462

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