Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions

Abstract

In this paper we establish weighted L q -L p -maximal regularity for linear vectorvalued parabolic initial-boundary value problems with inhomogeneous boundary conditions of static type. The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary. The novelty of the followed approach is the use of weighted anisotropic mixed-norm Banach space-valued function spaces of Sobolev, Bessel potential, Triebel-Lizorkin and Besov type, whose trace theory is also subject of study.

The heat equation with rough boundary conditions and holomorphic functional calculus

“…In [21] actually the general setting of higher order operators A with boundary conditions of Lopatinskii-Shapiro was consider. In [53] the first author extended the latter result to the weighted situation with γ ∈ (−1, p − 1), in which case δ ∈ ( 1 2 , 1) can only be taken arbitrarily close to 1 2 by taking γ close to p − 1. It would be interesting to investigate if one can extend special cases of [53] to other values of γ.…”

“…In [53] the first author extended the latter result to the weighted situation with γ ∈ (−1, p − 1), in which case δ ∈ ( 1 2 , 1) can only be taken arbitrarily close to 1 2 by taking γ close to p − 1. It would be interesting to investigate if one can extend special cases of [53] to other values of γ. In ours proofs the main technical reason that we can extend the range of γ's in the Dirichlet setting is that the heat kernel on a half space has a zero of order one at the boundary.…”

“…Details on traces in weighted Sobolev spaces can be found in [40] and [53]. We will need some simple existence results in one dimension.…”

“…By a standard localization argument it suffices to considerthe case O = R d + . The case γ ∈ (−1, p−1)is already considered in [53, Theorem 2.1 & Corollary 4.9] (also see[53, Theorem 4.4]), so from now on we will assume γ ∈ (p − 1, 2p − 1).Let us writeM := W ℓ,q (R, v; L p (R d + , w γ ; X)) ∩ L q (R, v; W k,p (R d + , w γ ; X)) v; L p (R d−1 ; X)) ∩ L q (R, v; B k− 1+γ p p,p (R d−1 ; X)). By Theorem 3.18, Proposition 7.3, Corollary 3.4 and [56, Propositions 5.5 & 5.6],M ֒→ H ℓ(1− 1 k ),q (R; v; [L p (R d + , w γ ; X), W k,p (R d + , w γ ; X)] 1 k ) ֒→ H ℓ(1− 1 k ),q (R; v; [H −1,p (R d + , w γ−p ; X), W k−1,p (R d + , w γ−p ; X)] 1 k ) = H ℓ(1− 1 k ),q (R; v; L p (R d + , w γ−p ; X)).Therefore, once applying Corollary 3.4,M ֒→ H ℓ(1− 1 k ),q (R; v; L p (R d + , w γ−p ; X)) ∩ L q (R, v; W k−1,p (R d + , w γ−p ; X)),(7.6) which reduces the problem to the A p -weight setting.…”

“…A combination of (7.10) and Lemma 7.13 gives the desired statement for the case O = R d + , from the general case follows by a standard localization argument. In order to show that there is a coretraction corresponding to Tr t=0 , it suffices to consider the case O = R d + and J = R. The case γ ∈ (−1, p − 1) follows from [53,Equation (38)], and therefore it remains to consider γ ∈ (p − 1, 2p − 1).…”