$H^{\infty }$-calculus for the Dirichlet Laplacian on conical domains

Abstract

We establish boundedness of the $H^{\infty }$-calculus for the Dirichlet Laplacian on conical domains in ${\mathbb{R}}^d$ and corresponding wedges on $L^p$-spaces with mixed weights. The weights are based on both the distance to the boundary and the distance to the tip/edge of the cone/wedge. Our main motivation comes from the study of stochastic partial differential equations and associated degenerate deterministic parabolic equations on non-smooth domains. As a consequence of our analysis, we also obtain maximal $L^p$-regularity for the Poisson equation on conical domains in appropriate weighted Sobolev spaces.

arXiv comment40 pages, 3 Figures, generalised from two-dimensional angular domains to cones of arbitrary dimension