Shape Design for Degenerate Parabolic Equations with Degenerate Boundaries and Its Application to Boundary Observability

TL;DR

This paper develops a shape design framework for degenerate parabolic equations with degenerate boundaries, establishing well-posedness, approximating solutions, and deriving boundary observability inequalities.

Contribution

It introduces a novel shape design method to approximate degenerate parabolic equations and applies it to obtain boundary observability results.

Findings

Established well-posedness of degenerate parabolic equations under Dirichlet conditions.

Proposed a shape design approach to approximate degenerate equations with uniform parabolic equations.

Derived a boundary observability inequality for the degenerate parabolic equation.

Abstract

In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate parabolic equation through a series of uniformly parabolic equations. Finally, as a tangible application of this shape design approach, we deduce a boundary observability inequality associated with the degenerate parabolic equation.