Concentrated terms and varying domains in elliptic equations: Lipschitz case

2014/12/18 by Gleiciane S. Aragão, G. S. Aragão, Aragão, G. S. +3 · 1 citation
Computer Science · Engineering · Mathematics · #34B15 #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations #math.AP #msc:34B15 #msc:35J61

paper · pdf · doi:10.48550/arxiv.1412.5850

13 pages, 2 figures

arxiv created 2014/12/18 · openalex publication_date 2014/12/18 · arxiv updated 2014/12/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H¹ , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.

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