Asymptotics in the Dirichlet Problem for Second Order Elliptic Equations with Degeneration on the Boundary

2021/06/30 by Mark Freidlin, Freidlin, Mark, Leonid Koralov +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR

paper · pdf · doi:10.48550/arxiv.2106.15766

arxiv created 2021/06/30 · openalex publication_date 2021/06/30 · arxiv updated 2021/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study small perturbations of the Dirichlet problems for second order elliptic equations that degenerate on the boundary. The limit of the solution, as the perturbation tends to zero, is calculated. The result is based on a certain asymptotic self-similarity near the boundary, which holds in the generic case. In the last section, we briefly consider the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of the solution depends on the time scale. Initial-boundary value problem with the Neumann boundary condition is discussed in the last section as well.

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