Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

Abstract

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted L q -maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A ∞class. In Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the A p -range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

Boundary Value Problems of Elliptic and Parabolic Type with Boundary Data of Negative Regularity

“…Maximal regularity results for the heat equation with inhomogeneous boundary data have been obtained in [27]. In [19] similar results were shown for general elliptic and parabolic boundary value problems. The elliptic and parabolic equations we are interested in are of the form…”

“…Proof. A similar proof was carried out in [19,Proposition 4.21]. We combine this proof with [21,Theorem 8.5.21] in order to obtain the R-bounded version.…”

“…This can be shown by induction on |α|. Using Lemma 4.2, the proof of [19,Lemma 4.22] carries over to our setting.…”

“…(5-2) holds for all α ∈ B(E) and all (s 1 , s 2 , s 3 ) ∈ R 3 . Note that the authors of [19] use a different convention concerning the sign of A. Taking (s 1 , s 2 , s 3 ) = (0, 1, 0) shows together with the iterated R-bounded versions of Mihklin's theorem, Proposition 3.8, show that…”

“…This way, we will not only be able to give a meaning to tr ∂O B j (D)u, but we will also obtain tools which help us to analyze the singularities of solutions at the boundary. This supplements the quantitative analysis in [19,27,29]. The idea to use spaces with mixed smoothness is quite essential in this paper, even if one refrains from using mixed scales.…”