On solutions to a class of degenerate equations with the Grushin operator

TL;DR

This paper investigates solutions to a class of degenerate elliptic equations involving the Grushin operator, establishing their asymptotic behavior near degenerate points and proving strong unique continuation properties.

Contribution

It introduces an Almgren-type monotonicity formula to precisely characterize the blow-up profiles of solutions at degenerate points.

Findings

Identified the exact asymptotic blow-up profiles of solutions.

Proved strong unique continuation properties for solutions.

Extended understanding of degenerate elliptic equations with the Grushin operator.

Abstract

The Grushin Laplacian $- \Delta_\alpha $ is a degenerate elliptic operator in $\mathbb{R}^{h+k}$ that degenerates on $\{0\} \times \mathbb{R}^k$. We consider weak solutions of $- \Delta_\alpha u= Vu$ in an open bounded connected domain $\Omega$ with $V \in W^{1,\sigma}(\Omega)$ and $\sigma > Q/2$, where $Q = h + (1+\alpha)k$ is the so-called homogeneous dimension of $\mathbb{R}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $\Omega$. As an application we derive strong unique continuation properties for solutions.