Gradient estimates for higher order elliptic equations on nonsmooth domains

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

Abstract. This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for t...

Carleman estimates and unique continuation for second order elliptic equations with nonsmooth coefficients

Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains

We establish the global Hölder estimates for solutions to second-order elliptic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an exterior cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on...

Higher Order Derivative Estimates for Finite-difference Schemes for Linear Elliptic and Parabolic Equations

Singular Sets of Higher Order Elliptic Equations

The implicit function theorem implies that the zero set of a smooth function, the set where the function vanishes, is a smooth hypersurface away from the critical zero set. Hence to study zero sets it is important to understand the structure of the critical zero sets. For solutions of the second order elliptic equations the critical zero sets represent the singular parts of zero sets. They have...