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

TL;DR

This paper investigates the asymptotic behavior of solutions to second order elliptic equations with boundary degeneracy, analyzing perturbations, boundary conditions, and related parabolic equations to understand solution limits and metastability effects.

Contribution

It introduces a novel asymptotic analysis method for degenerate elliptic equations near the boundary, including boundary condition variations and parabolic extensions.

Findings

Solution limits are explicitly calculated as perturbations tend to zero.

Asymptotic self-similarity near the boundary is established in the generic case.

Metastability effects are identified in the parabolic case depending on time scale.

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.