BOUNDARY BEHAVIOR OF THE LAYER POTENTIALS FOR THE TIME FRACTIONAL DIFFUSION EQUATION IN LIPSCHITZ DOMAINS

摘要

This paper investigates the boundary behavior of layer potentials for the time fractional diffusion equation ( TFDE) in Lipschitz domains. The paper is a continuation of [9]. Since now the boundary of the spatial domain Ω admits only Lipschitz smoothness, we have to replace the classical technique used in [9] with a more delicate harmonic analysis technique. We prove that certain nontangential maximal functions related to the layer potentials are bounded in L~p(Σ_T), which in particular implies the usual jump relations known for the heat equation. Although the results are well known in the case of the heat potential corresponding to the case a = 1, the proofs of the same properties seem not to be available in the case 0 < a < 1.