Mixed type boundary value problem of elliptic equation in a thin domain

TL;DR

This paper establishes uniform a priori estimates for a class of elliptic boundary value problems in thin, crescent-shaped domains with mixed boundary conditions, providing insights into their asymptotic behavior as the domain collapses.

Contribution

It introduces novel uniform Schauder estimates for elliptic equations in thin domains with mixed boundary conditions, extending existing theory to narrow, crescent-shaped regions.

Findings

Proved uniform Schauder estimates in thin crescent-shaped domains.

Derived asymptotic estimates as the domain collapses into a segment.

Established boundary behavior under mixed boundary conditions.

Abstract

In this paper, we prove the a priori estimates for two-dimensional second order homogeneous linear elliptic equations in a narrow region. In a crescent-shaped area, part of the boundary is subject to an oblique derivative boundary condition, while the other part of the boundary is subject to a Dirichlet boundary condition. We show that, as the crescent-shaped area collapses into a segment under suitable conditions, the boundary value problem obeys uniform Schauder estimates and induces an asymptotic estimate.