In this paper, we investigate linear elliptic, second-order boundary value problems with mixed boundary conditions on domains with a rough boundary. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove first Hölder continuity of the solution. Secondly, Gaussian Hölder estimates for the corresponding heat kernel are derived.
"Hölder estimates for second-order operators on domains with rough boundary." Adv. Differential Equations 20 (3/4) 299 - 360, March/April 2015.