Differential operators on domains with conical points: precise uniform regularity estimates

2016/05/24 by Constantin Bǎcuţǎ, Bacuta, Constantin, Hengguang Li +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1605.07907

openalex publication_date 2016/05/25 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28

Abstract

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated Poisson problem with mixed boundary conditions. The coefficients and the solutions belong to (suitable) weighted Sobolev spaces. The space of coefficients is a Banach space that contains, in particular, the space of smooth functions. Hence, our results extend classical well-posedness results for strongly elliptic equations in domains with conical points to problems with singular coefficients. We furthermore provide precise uniform estimates on the norms of the solution operators.

Related

• Differential Operators on Conic Manifolds: Maximal Regularity and Parabolic Equations
• Analysis and boundary value problems on singular domains: an approach via bounded geometry
• Gradient regularity for strongly singular or degenerate elliptic and parabolic equations
• Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations
• H^∞-calculus for the Dirichlet Laplacian on conical domains