2025/05/16 by Gabriele Cora, Cora, Gabriele, Gabriele Fioravanti +3
Mathematics · #35B07 #35B40 #35B65 #35J70 #35J75 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2505.16534
openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Lau(z)=div(|y|a∇ u)(z), where z=(x,y)∈\mathbb Rd-n×\mathbb Rn, 2≤ n≤ d are two integers and a∈\mathbb R. The weight term is degenerate/singular on the (possibly very) thin characteristic manifold Σ0=\|y|=0\ of dimension 0≤ d-n≤ d-2. Whenever a+n>0, we prove smoothness of the axially symmetric La-harmonic functions. In the mid-range a+n∈(0,2), we deal with regularity estimates for solutions with inhomogeneous conormal boundary conditions prescribed at Σ0, 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 Σ0 by a boundary Harnack type principle.