Remarks on elliptic equations degenerating on lower dimensional manifolds
Gabriele Cora, Gabriele Fioravanti, Stefano Vita
TL;DR
The paper studies elliptic equations with coefficients that degenerate on a lower-dimensional manifold, modeled by $L_a=\\operatorname{div}(|y|^a\\nabla \cdot)$ with thin set $\\Sigma_0=\\{|y|=0\\}$ and codimension $n$. It develops a unified regularity theory across regimes $a+n>0$, $(0,2)$, and $a+n<2$, notably proving local $C^{\infty}$-regularity for axially symmetric $L_a$-harmonic functions when $a+n>0$, and establishing inhomogeneous conormal boundary results and a Dirichlet-to-Neumann link to fractional Laplacians in the mid-range. It then constructs a higher-codimension extension theory, including a trace from $\\mathcal{D}^{1,a}(\\mathbb{R}^d)$ to $\\mathcal{D}^s(\\mathbb{R}^{d-n})$ with $s=(2-a-n)/2$, and an explicit extension $U=u*P$ using a Poisson kernel $P$ that realizes $(-\\Delta)^s$ on $\\Sigma_0$ as a Dirichlet-to-Neumann map. In the homogeneous Dirichlet setting for $a+n<2$, the authors derive Hölder and Schauder regularity via a boundary Harnack principle, yielding sharp $C^{2-a-n}$ regularity under codimension assumptions and Lipschitz regularity in the endpoint case $a+n=1$, $n\ge 4$. Overall, the work extends prior results and provides a cohesive framework linking degenerate elliptic equations on thin manifolds with higher-codimension fractional operators.
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 $Σ_0=\{|y|=0\}$ 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 $Σ_0$, 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 $Σ_0$ by a boundary Harnack type principle.