Differential operators on domains with conical points: precise uniform regularity estimates
Provides rigorous functional analytic foundations for elliptic PDEs with singular coefficients on non-smooth domains, benefiting analysts working on boundary value problems.
The paper establishes uniform regularity estimates for strongly elliptic second-order differential operators with singular coefficients on domains with conical points, extending classical well-posedness results to singular coefficients. Precise uniform bounds on solution operator norms are provided.
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.