E-Books →Elliptic Theory in Domains with Boundaries of Mixed Dimension

Take an open domain Ω⊂Rn whose boundary may be composed of pieces of different dimensions. For instance, Ω can be a ball on R3, minus one of its diameters D, or a so-called saw-tooth domain, with a boundary consisting of pieces of 1-dimensional curves intercepted by 2-dimensional spheres. It could also be a domain with a fractal (or partially fractal) boundary. Under appropriate geometric assumptions, essentially the existence of doubling measures on Ω and ∂Ω with appropriate size conditions. The authors construct a class of second order degenerate elliptic operators L adapted to the geometry, and establish key estimates of elliptic theory associated to those operators. This includes boundary Poincaré and Harnack inequalities, maximum principle, and Hölder continuity of solutions at the boundary.