A Shape Design Approximation for Degenerate Partial Differential Equations and Its Application

TL;DR

This paper introduces a novel shape design approximation method for degenerate PDEs, enabling new Carleman estimates and null controllability results without requiring second order derivatives.

Contribution

The paper presents a new approximation approach for degenerate PDEs that simplifies deriving Carleman estimates and controllability results, avoiding second order derivative constraints.

Findings

Derived a Carleman estimate for backward degenerate parabolic equations.

Established null controllability for degenerate parabolic equations using the new method.

Enabled analysis without second order derivatives in degenerate equations.

Abstract

In this paper, we focus on two types of degenerate partial differential equations: a degenerate elliptic equation and a degenerate parabolic equation. Significantly, both categories are characterized by the same principal operator. To obtain solutions for these equations, we introduce a novel approximation approach, termed the shape design approximation. As a practical application of this method, we derive a Carleman estimate for the backward degenerate parabolic equation. This estimate plays a pivotal role in establishing the null controllability of the degenerate parabolic equation. A notable advantage of employing the shape design approximation in deriving the Carleman estimate is that it enables us to bypass the requirement for second order derivatives in the degenerate equation. Usually, this has been a significant obstacle in the derivation of Carleman estimates for degenerate parabolic equations.