Remarks on elliptic equations degenerating on lower dimensional manifolds

TL;DR

This paper advances the understanding of regularity for elliptic equations with coefficients degenerating on lower-dimensional manifolds, connecting to fractional Laplacians and extending previous results.

Contribution

It provides new regularity results for degenerate elliptic equations, especially relating to fractional Laplacians and boundary conditions on thin manifolds.

Findings

Smoothness of axially symmetric solutions when a+n>0

Regularity estimates for solutions with boundary conditions when a+n in (0,2)

Boundary Harnack principle for solutions with Dirichlet conditions when a+n<2

Abstract

The paper continues the analysis started in [Cora-Fioravanti-Vita-25,Fioravanti-24] on the local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold. The model operator is given by $L_au(z)=\mathrm{div}(|y|^a\nabla u)(z)$, where $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, $2\leq n\leq d$ are two integers and $a\in\mathbb R$. The weight term is degenerate/singular on the (possibly very) thin characteristic manifold $\Sigma_0=\{|y|=0\}$ of dimension $0\leq d-n\leq d-2$. Whenever $a+n>0$, we prove smoothness of the axially symmetric $L_a$-harmonic functions. In the mid-range $a+n\in(0,2)$, we deal with regularity estimates for solutions with inhomogeneous conormal boundary conditions prescribed at $\Sigma_0$, and we establish the connection with fractional Laplacians on very thin flat manifolds via Dirichlet-to-Neumann maps, as a higher codimensional analogue of the extension theory developed by Caffarelli and Silvestre. Finally, whenever $a+n<2$ we complement the study in [Fioravanti-24], providing some regularity estimates for solutions having a homogeneous Dirichlet boundary condition prescribed at $\Sigma_0$ by a boundary Harnack type principle.