On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

Abstract

The goal of the work is to verify the fractional Leibniz rule for Dirichlet Laplacian in the exterior domain of a compact set. The key point is the proof of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

The Leibniz rule for the Dirichlet and the Neumann Laplacian

“…As for the low spectrum, which is essetial for the homogeneous spaces, it depends on domains. The bounded domain case has no restriction, but the possible regularity in the exterior domain case is restricted to smaller range because of the slower decay of gradient estimates for the heat kernel (see papers [7,8]).…”

“…In general domains, we refer to a paper [14] which studies the bilinear estimates in Besov spaces associated with the Dirichlet Laplacian with the regularity 0 < s < 2 by means of the gradient estimates for the heat equation in L p . The exterior domainn case is discussed in a paper [7]. We also refer to several papers by Di Nezza, Palatucci and Valdinoci [3], and Tartar [21] for fractional Sobolev spaces on domains.…”