Approximation of Elliptic Equations with Interior Single-Point Degeneracy and Its Application to Weak Unique Continuation Property

TL;DR

This paper develops an approximation approach to analyze elliptic equations with interior degeneracy, establishing well-posedness, a three-ball theorem, and proving the quantitative weak unique continuation property in high dimensions.

Contribution

It introduces a novel approximation method to handle interior degeneracy in elliptic equations and proves QWUCP using the three-ball theorem.

Findings

Established well-posedness in weighted spaces

Derived the three-ball theorem at degenerate points

Proved QWUCP for specific cases

Abstract

This paper investigates the quantitative weak unique continuation property (QWUCP) for a class of high-dimensional elliptic equations with interior point degeneracy. First, we establish well-posedness results in weighted function spaces. Then, using an innovative approximation method, we derive the three-ball theorem at the degenerate point. Finally, we apply the three-ball theorem to prove QWUCP for two different cases.