Remarks on elliptic equations degenerating on lower dimensional manifolds

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 $L_{a}u(z)=\mathrm{div}(|y{|}^a\nabla u)(z)$, where $z=(x,y)\in {\mathbb{R}}^{d-n}\times {\mathbb{R}}^n$, $2\le n\le d$ 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 $0\le d-n\le d-2$. 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.

13 pages. This paper was originally part of the paper "Schauder estimates for elliptic equations degenerating on lower dimensional manifolds", arXiv:2501.19033