2011/04/08 by Tilak Bhattacharya, Bhattacharya, Tilak, Ahmed Mohammed +1 · 2 citations
Computer Science · Mathematics · #35J60 #35J70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #msc:35J60 #msc:35J70
paper · pdf · doi:10.48550/arxiv.1104.1462
34 pages; Minor revisions made
openalex publication_date 2011/04/08 · arxiv created 2011/06/28 · arxiv updated 2011/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem Δ_∞ 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 \citeBMO on this topic.