Besov regularity of inhomogeneous parabolic PDEs

We study the regularity of solutions of parabolic partial differential equations with inhomogeneous boundary conditions on polyhedral domains D⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subset \mathbb {R}^3$$\end{document} in the specific scale Bτ,τα,1τ=α3+1p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ B^{\alpha }_{\tau ,\tau }, \ \frac{1}{\tau }=\frac{\alpha }{3}+\frac{1}{p}\ $$\end{document} of Besov spaces. The regularity of the solution in this scale determines the order of approximation that can be achieved by adaptive numerical schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our results are in good agreement with the forerunner (Dahlke and Schneider in Anal Appl 17:235–291, 2019), where parabolic equations with homogeneous boundary conditions were investigated.


Introduction
This paper is concerned with the regularity theory of partial differential equations (=PDEs) of parabolic type. The analysis of parabolic PDEs has a long history and a lot of results concerning existence and uniqueness of solutions are known. However, in most cases, an analytic expression of the solution is not available, so that numerical algorithms for the constructive approximation of the unknown solution up to a given tolerance are needed. Our main goal here is to justify the use of adaptive schemes when solving parabolic PDEs numerically, in This article is part of the section "Theory of PDEs" edited by Eduardo Teixeira. particular on non-smooth domains ⊂ R d . As a prototype of our investigations one may think of the heat equation with inhomogeneous initial boundary conditions, i.e.,

∂u ∂t
− u = f in (0, T ] × , u = g on [0, T ] × ∂ , u| t=0 = u 0 in , where u is the unknown solution describing the distribution of heat over time, the right hand side f represents an outer heat source, g is some adequate function defined on the boundary, and u 0 is a function which represents the initial temperature at time t = 0. We treat more general parabolic PDEs of higher order in the course of our investigations as well. We are interested in regularity properties of the unknown solution u in specific nonstandard smoothness spaces, i.e., the so-called adaptivity scale of Besov spaces B α τ,τ ( ), where α > 0 stands for the smoothness of the solution and τ displays its integrability. The smoothness in these scales of spaces is related with the achievable convergence order of adaptive algorithms. To briefly explain the concept, in an adaptive strategy the choice of the underlying degrees of freedom is not a priori fixed but depends on the shape of the unknown solution. In particular, additional degrees of freedom are only spent in regions where the numerical approximation is still 'far away' from the exact solution.
On the other hand, it is the regularity of the solution in the scale of (fractional) Sobolev spaces H s ( ), where now s indicates the smoothness of u, which encodes information on the convergence order for nonadaptive (uniform) methods. Thus, we can justify the use of adaptivity if the smoothness of the exact solution of a given PDE within the scale of Besov spaces is high enough compared to its classical Sobolev smoothness, i.e., when α > s.
For elliptic PDEs there are a lot of positive results in this context. For example, for the Poisson equation with homogeneous Dirichlet conditions it is well-known that if the domain under consideration, the right-hand side, and the coefficients are sufficiently smooth, then the problem is completely regular and there is no reason why the Besov smoothness should be higher than the Sobolev smoothness. However, on nonsmooth (e.g. polyhedral) domains, the situation changes completely. On these domains, singularities at the boundary may occur that diminish the Sobolev regularity of the solution significantly, cf. [6,17,18,20] (but can be compensated by suitable weight functions). However, recent studies show that these boundary singularities do not influence the Besov regularity too much, cf. [7,8], so that for certain nonsmooth domains the use of adaptive algorithms is completely justified. In [29] the authors contributed to these studies and also obtained positive results for the Poisson equation with inhomogeneous (Dirichlet, Neumann as well as mixed) boundary conditions.
Concerning parabolic PDEs not much is known in this respect so far. A first step in this direction was done by Aimar et al. [1] and later on continued in [2,3]. These studies are limited to the heat equation with zero Dirichlet boundary conditions, i.e., (1.0.1) with g = u 0 = 0. Moreover, in [10,11] the authors obtained Besov regularity estimates for more general linear and nonlinear parabolic PDEs but again for zero Dirichlet boundary conditions only. In good agreement with the elliptic setting in all parabolic problems studied it turns out that (again) adaptivity pays off.
Unfortunately, all results established so far for parabolic PDEs do not deal with inhomogeneous boundary conditions, which is very restrictive in view of possible applications. It is our intention to close this gap now with our investigations.
As a main tool for establishing our results we work with weighted Sobolev spaces defined on domains of polyhedral type D, whose importance in this context comes from the fact that the weights can compensate the above mentioned singularities of the solutions at the boundary. Our specific weighted Sobolev spaces (which will be briefly refered to as Vspaces in the sequel) are taken from Maz'ya, Rossmann [26] and can be seen as a refinement of the well-known Kondratiev spaces. However, in contrast to the Kondratiev spaces, our V -spaces here invoke mixed weights, which reflect in a more subtle way the distance(s) to the vertices and edges of the underlying polyhedral domain.
Kondratiev spaces have been heavily used for regularity studies in the past, since they are very much related with Besov spaces in the adaptivity scale (1.0.2), in the sense that powerful embedding results exist, see, e.g. [19].
For our purposes, in Theorem 2.10 we transfer these embeddings to the setting of our V -spaces and obtain subject to some further restrictions on the parameters. In particular, the parameter l above denotes the smoothness and p the integrability of the function considered, whereas β, δ are exponents related with the weights measuring the distance to the vertices and edges of the domain D of polyhedral type. As can be seen from (1.0.3), having established regularity in the scale of V -spaces as well as fractional Sobolev spaces (using the coincidence B s 2,2 (D) = H s (D)) in turn yields Besov regularity of the solution under consideration.
Having this in mind we process as follows: first we improve and generalize the findings from [10,28] for parabolic PDEs with homogeneous boundary conditions from Kondratiev to V -spaces.
Afterwards, in order to obtain regularity results for parabolic PDEs with inhomogeneous initial boundary conditions, we adapt the proofs from [28] by reducing the inhomogenous parabolic problem to a similar (easier) one, where only homogeneous boundary terms appear (e.g. for the heat equation in (1.0.1) this leads to a similar problem with new boundary termg = 0, a new right hand sidef and initial conditionũ 0 , which then depend on g). There are similar strategies to reduce inhomogeneous elliptic problems to homogeneous ones, but for parabolic equations this is technically more involved and requires a more subtle analysis (in particular in view of the regularity of g). With this reduction strategy we are then able to establish Sobolev and weighted Sobolev regularity results for parabolic PDEs with inhomogeneous boundary conditions. These together with our embedding result (1.0.3) finally allow us to establish Besov and Hölder-Besov regularity for our parabolic problems.
The paper is organized as follows. In Sect. 2 we introduce the relevant function spaces needed for our investigations, in particular, the weighted Sobolev spaces (V -spaces), Besov spaces, and certain Banach-space valued function spaces as well as Hölder-spaces.
In Sect. 3 we provide regularity results in Sobolev spaces with and without time derivatives for the inhomogeneous parabolic problem. Afterwards we establish regularity results the scale of V -spaces for the homogeneous as well as the inhomogeneous problem. Then using our embedding result we deduce regularity in the scales of (Hölder-)Besov spaces. Finally, we give a short discussion and comparison of our findings with the forerunners from [28].

Notation
We start by collecting some general notation used throughout the paper. As usual, N stands for the set of all natural numbers, N 0 = N ∪ {0}, Z denotes the integers, and R d , d ∈ N, is the d-dimensional real Euclidean space with |x|, for x ∈ R d , denoting the Euclidean norm of x. Moreover, by [·] we denote the floor function [x] := max{m ∈ Z : m ≤ x} for x ∈ R. Let N d 0 be the set of all multi-indices, α = (α 1 , . . . , α d ) with α j ∈ N 0 and |α| := d j=1 α j . Then for partial derivatives we write is the open ball of radius r > 0 centered at x.
We denote by c a generic positive constant which is independent of the main parameters, but its value may change from line to line. The expression A B means that A ≤ c B. If A B and B A, then we write A ∼ B.
Throughout the paper 'domain' always stands for an open and connected set. The test functions on a domain are denoted by C ∞ 0 ( ). Let L p ( ), 1 ≤ p ≤ ∞, be the Lebesgue spaces on as usual. We denote by C( ) the space of all bounded continuous functions f : → R and C k ( ), k ∈ N 0 , is the space of all functions f ∈ C( ) such that ∂ α f ∈ C( ) for all α ∈ N 0 with |α| ≤ k, endowed with the norm |α|≤k sup x∈ |∂ α f (x)|.
Moreover, the set of distributions on will be denoted by D ( ), whereas S (R d ) denotes the set of tempered distributions on R d .

Lipschitz diffeomorphism and Lipschitz domain
We say that the one-to-one mapping : R d → R d is a Lipschitz diffeomorphism, if the components k (x) of (x) = ( 1 (x), . . . , d (x)) are Lipschitz functions on R d and where the equivalence constants are independent of x and y. Then the inverse of is also a Lipschitz diffeomorphism on R d . We define a Lipschitz domain according to [28,Def. 2.1.1] as follows: Let be a bounded domain in R d with boundary = ∂ . Then is said to be a (bounded) Lipschitz domain, if there exist N open balls K 1 , . . . , K N such that N j=1 K j ⊃ and K j ∩ = ∅ if j = 1, . . . , N , with the following property: for every ball K j there are Lipschitz diffeomorphisms ψ ( j) such that ψ ( j) :

Sobolev and fractional Sobolev spaces
Let m ∈ N 0 and 1 ≤ p ≤ ∞. Then W m p ( ) denotes the standard L p -Sobolev spaces of order m on the domain , equipped with the norm where D( ) stands for the collection of all infinitely differentiable functions with support compactly contained in .
If p = 2 we shall also write H m ( ) instead of W m 2 ( ). Moreover, for s ∈ R we define the fractional Sobolev spaces H s (R d ) as the collection of all u ∈ S (R d ) such that where F denotes the Fourier transform with inverse F −1 . These spaces partially coincide with the classical Sobolev spaces for s = m with m ∈ N 0 . Corresponding spaces on domains can be defined via restrictions of functions from H s (R d ) equipped with the norm If, additionally, ⊂ R d is a bounded Lipschitz domain, then for s = m + σ with m ∈ N 0 and σ ∈ (0, 1), an alternative norm comes from the Sobolev-Slobodeckij spaces and is given by  (1) , . . . , x (d ) , (ii) for every ξ ∈ M k there exist a neighborhood U ξ and a diffeomorphism (a C ∞ mapping) κ ξ which maps D ∩ U ξ onto D ξ ∩ B 1 , where D ξ is a dihedron and B 1 is the unit ball, (iii) for every vertex x ( j) there exist a neighborhood U j and a diffeomorphism κ j mapping D ∩ U j onto K j ∩ B 1 , where K j is a cone with edges and vertex at the origin.

Remark 2.2 •
We do not exclude the cases d = 0 and d = 0. In the latter case, the set S consists only of smooth non-intersecting edges. In Fig. 1 (1) , . . . , x (d ) . We denote the distance of x from the edge M k by r k (x) and the distance from the vertex x ( j) by ρ j (x). Furthermore, we denote by X j the set of all indices k such that the vertex x ( j) is an end point of the edge M k . Let U 1 , . . . , U d be domains in R 3 such that . This space is provided with the norm The corresponding norm is defined as We provide embeddings between the spaces V l, p β,δ (D), which will be needed later on when establishing our regularity results. First, we recall [26,Lem. 4.1.3]. Lemma 2.5 Let l, l be nonnegative integers and 1 < p ≤ q < ∞ such that l − 3/ p ≥ l − 3/q. Then we have the embedding

Besov spaces via wavelets
Under certain restrictions on the parameters, the Besov spaces B s p,q , defined by the classical approach via higher order differences or alternatively by the Fourier-analytical approach coincide and even allow an equivalent characterization in terms of wavelet decompositions, see e.g. [33,34] and also [5,25]. Therefore, in this section we introduce Besov spaces via wavelet decompositions as in [19,Sec. 2,p. 563], since this characterization is extremely convenient for our purposes when providing embedding results between weighted Sobolev and Besov spaces later on. Consider the wavelet construction of Daubechies, see e.g. [13,14], with the mother function η and the scaling function φ, which satisfy the following conditions: We set ψ 0 = φ, ψ 1 = η, define E as the set of the nontrivial vertices of [0, 1] d , and put Moreover, let := {ψ e : e ∈ E} and consider the set of dyadic cubes, Now we put = D + × , where D + is the set of the dyadic cubes with measure at most 1. Furthermore, we use the notation D j := {I ∈ D : |I | = 2 − jd }. The functions from are rescaled in the following way: Then the set {ψ I : I ∈ D, ψ ∈ } forms an orthonormal basis in L 2 (R d ). Let Q(I ) denote some dyadic cube of minimal size such that supp ψ I ⊂ Q(I ) for every ψ ∈ . Then we can write every function f ∈ L 2 (R d ) as where P 0 is the orthogonal projector onto the closure of the span of the function (x) = φ(x 1 ) · · · φ(x d ) and its integer shifts (· − k), k ∈ Z d , in L 2 (R d ). It will be convenient to include φ into the set . We use the notation φ I := 0 for |I | < 1, φ I = φ(· − k) for I = k + [0, 1] d , and can simply write With these considerations we can now define Besov spaces B s p,q (R d ) by decay properties of the wavelet coefficients, if the parameters fulfill certain conditions, as follows.
Choose r ∈ N such that r > s and construct a wavelet Riesz basis as described above. Then the function f ∈ L p (R d ) belongs to the Besov space B s p,q (R d ) if it admits a decomposition of the form Corresponding spaces on domains ⊂ R d can be defined by restriction via Using the wavelet characterization of the Besov spaces, in [29, Thm. 3.1, Sec. 3] we established an embedding result on polyhedral cones, which displays the close relation between the V -and Besov spaces, respectively. Applying localization arguments we now generalize this embedding for domains of polyhedral type, which will be one of our main tools when investigating the Besov regularity of solutions for the parabolic problems.

Proof
Step 1: The weighted Sobolev space V l, p β,δ (K ) can be seen as a subcase of Definition 2.1 for the bounded polyhedral cone K with edges M 1 , . . . , M d , since K is also a domain of polyhedral type, where now d = 1, β = β ∈ R, X 1 = {1, . . . , d} in the norm (2.1.1), and the integral domain D ∩ U 1 is now K itself with U 1 being the unit ball around the vertex of the cone.
Moreover, according to their definition, domains of polyhedral type D can be covered by i.e., if M k is not adjacent to x ( j) . Since the diffeomorphisms κ j in Definition 2.1 are C ∞ (and therefore also Lipschitz-diffeomorphisms), we have using the transformation theorem, where κ j (D ∩ U j ) is a bounded polyhedral cone according to Definition 2.3 and δ ( j) contains the components of δ related to the edges of the cone κ j (D ∩ U j ), i.e., the components with indices from X j .
Since the definition of the spaces V l, p β,δ (D) is independent of the covering U j , cf. Remark 2.4, we can take a slightly larger covering where ε > 0 is sufficiently small. Furthermore, we can take a subordinated family {ψ j } d j=1 of smooth functions satisfying for j = 1, . . . , d , and define a resolution of unity subject to U j as follows: The functions η j are well defined since d j=1 ψ j (x) is non-zero for all x ∈ U j , j = 1, . . . , d . Furthermore, since every point in D is covered at least once and at most d -times by U j , we have for x ∈ D that We now show that the following norms are equivalent: which shows (2.2.5). Step Taking the pth power on both sides and summing over j = 1, . . . , d together with (2.2.5) and (2.2.6) we obtain the desired result

Remark 2.11
We briefly discuss the role of β and δ, where for simplicity we assume that β has the same components, i.e., β = (β, . . . , β) for some β ∈ R. In particular, these two parameters "compensate" possible singularities at the vertices or at the edges of the domain of polyhedral type D with the help of the weight functions ρ j and r k , respectively. Moreover, in order to deal with L p functions we require that β j ≤ l for all j = 1, . . . , d and δ i ≤ l for all i = 1, . . . , d, cf. Remark 2.4 (the second condition is incorporated in the even more restrictive inequality 0 ≤ r < 3(l − |δ + |)). Moreover, by using [26,Lem. 4.1.3], we see that for all k = 1, . . . , d, which implies |δ| ≤ |δ |. Furthermore, the smaller |δ + |, the greater the smoothness parameter r < 3(l − |δ + |) can be chosen. The embedding (2.2.7) of the V -scale is reflected in the condition for r , since decreasing values of |δ + | correspond to shrinking V -spaces, which then in turn can be embedded into smaller Besov spaces, i.e. with larger smoothness parameters r (and hence a growing upper bound for r can be observed).
The dependence on β is slightly different. Also in this context from [26,Lem. 4 Therefore, in this case we see that for every (fixed) l we can choose any β < l with the effect that the resulting space V l, p β,δ (D) still embeds into the same Besov space with smoothness r < min{l, 3(l − |δ + |)} independent of β (again ignoring the dependence on s for the moment). Moreover, the weight parameter β (connected to the distances to the vertices of D) not playing any role in the condition on r (besides the requirement max j=1,...,d β j = β < l needed for the embedding into L p ) is analogous to previous results for smooth cones, cf. [12,Sec. 2], where the singular set is just a point (of dimension 0).

Banach-space valued function spaces
In order to investigate parabolic problems in the sequel we introduce some Banach-valued function spaces following [28, Subsec. 2.2.2].

Definition 2.12
Let X be a Banach space and I ⊂ R some interval. We denote by L p (I , X ), 1 ≤ p ≤ ∞, the space of (equivalence classes of) measurable functions u : I → X such that the mapping t → u(t) X belongs to L p (I ), which is endowed with the norm Furthermore, for m ∈ N 0 we denote by W m p (I , X ) the space of all functions u ∈ L p (I , X ), whose weak derivatives of order 0 ≤ k ≤ m belong to L p (I , X ), normed by In particular, for D T = (0, T ] × D with D being some domain of polyhedral type we abbreviate and and where the respective norms for the spaces considered in the second embedding are essentially the same. For later use we need also the following embedding theorem which is a direct consequence of the proof of [28,Thm. 2.4.12]. Theorem 2.14 Let X , X 1 , X 2 be Banach spaces such that X 1 ∩ X 2 → X . Moreover, let k ∈ N 0 , 0 < p ≤ ∞, and 0 < T < ∞. Then the following embedding holds In the sequel we shall need the following Banach-space valued Hölder spaces, which can be also found in [  Moreover, we say that u ∈ C k (I , X ), k ∈ N 0 , if u has a Taylor expansion is then equipped with the following norm Given α ∈ (0, 1), we denote by C α (I , X ) the Hölder space containing all u ∈ C(I , X ) such that Consequently, C k,α (I , X ) contains all functions u ∈ C(I , X ) such that

Remark 2.16 (i) We have the following embedding
This can be shown as follows: For real Hilbert spaces V , H such that Moreover, in a similar way we can show that For later use we provide the following auxiliary embedding result which is a generalization of [16, Thm. 4, § 5.9.2].

Lemma 2.17 Let D be an open bounded Lipschitz domain
and the following estimate holds where the constant C is independent of u.
Proof The proof is based on the proof in [16, Thm. 4, § 5.9.2] with suitable modifications according to our needs. We consider first m = 0, in which case and for an appropriate constant C independent of u. Moreover, since E is a universal extension operator, it is also an extension operator from L 2 (D) into L 2 (V ), and hence, by a similar argument as in [16, Thm. 4, § 5.9.2] we obtain that ∂ tū ∈ L 2 ([0, T ], L 2 (V )) with the estimate We proceed by assuming for the moment thatū is a smooth function. We then compute where we used Hölder's inequality, the inequality 2ab ≤ a 2 + b 2 , and the fact that there is no boundary term when we integrate by parts, since the extensionū has compact support within V . Integrating both sides of (2.3.10) w.r.t. the time-variable t from s 1 to s 2 for 0 ≤ s 1 ≤ s 2 ≤ T we obtain (2.3.11) Since (2.3.11) holds for arbitrary 0 ≤ s 1 ≤ s 2 ≤ T we choose them such that one of s 1 , s 2 provides the minimum of ū(·)|H k (V ) and the other one provides the maximum. We denote them by t max and t min , respectively. Then for the left-hand side of (2.3.11) we calculate w.l.o.g. for s 1 = t min , s 2 = t max that The case when u is not smooth follows by approximating u using a mollifier, i.e., u ε = η ε * u, as in in [16,Thm. 4, § 5.9.2]. As above it follows in this case that u ∈ C([0, T ], H k (D)).
It remains to show the case when m ≥ 1, which can be reduced to the case when m = 0 as follows. Let α be a multiindex of order |α| ≤ m and set v α : Hence, we can apply (2.3.15), with u replaced by v α , and sum over all indices |α| ≤ m which leads to the desired estimate (2.3.7). This completes the proof.

The fundamental parabolic problems
We present the fundamental parabolic problems with homogeneous and inhomogeneous boundary conditions, that we want to consider in the sequel. Let D ⊂ R 3 be some domain of polyhedral type with faces j , j = 1, . . . , n.
Parabolic problem with homogeneous boundary conditions. Let m ∈ N. We consider the following initial-boundary value problem, Here f is a function given on D T , ν denotes the exterior normal to j , and the partial differential operator L is given by where a αβ are bounded real-valued functions from C ∞ (D T ) with a αβ = a βα . Furthermore, the operator L is assumed to be uniformly elliptic with respect to t ∈ [0, T ], i.e., |α|,|β|=m Let us denote by the time-dependent bilinear form. In the sequel, w.l.o.g. we suppose that B(t, ·, ·) satisfies for all u ∈H m (D) and a.e. t ∈ [0, T ]. Moreover, we set Finally, we explain what is meant by the weak solution of problem (P hom ).
is called a weak solution of problem (P hom ) if, and only if, u(0, ·) = 0 in D and the equality It is also our intention to study inhomogeneous versions of (P hom ). Therefore, we modify (P hom ) as follows.
Parabolic problem with inhomogeneous boundary conditions. Let m ∈ N. We consider the following initial-boundary value problem, where g and u 0 are functions given on D T and D, respectively, and f , L are as explained before for problem (P hom ). We wish to point out here that the condition on the coefficients a αβ of L to be smooth is quite strong and can probably be weakened. This requires a careful check of the subsequent regularity results. However, the since presentation is already quite technical and not easy to read, we decided to merely focus on the (necessary) regularity assumptions regarding the right hand side f as well as the initial and boundary conditions u 0 and g, respectively.

Remark 2.19
In analogy to Definition 2.18 we can define what is meant by a weak solution of the inhomogeneous problem (P inhom ) as follows. (2.4.5) and the equality

Operator pencils
for j = 1, . . . , n and k = 1, . . . , m, where D ⊂ R 3 is a domain of polyhedral type with faces j . We assume that is a uniformly elliptic differential operator of order 2m with smooth coefficients A α .

Operator pencils
The boundary value problem (2.5.1) generates two types of operator pencils, A ξ (λ) and A j (λ), for the edge points and vertices of D, respectively.

Operator pencils A ξ (λ) for edges:
Let ξ be a point on an edge M k , and let k + , k − be the faces of D adjacent to ξ . Then by D ξ we denote the dihedron which is bounded by the half-planes˚ k ± tangent to k ± at ξ and the edgeM ξ =˚ k + ∩˚ k − . Furthermore, let r , ϕ be polar coordinates in the plane perpendicular toM ξ such that Then we define the operator A ξ (λ) as follows: denotes the principal part of the differential operator L(x, ∂ x ) with coefficients frozen at ξ . The operator A ξ (λ) realizes a continuous mapping for every λ ∈ C, where I ξ denotes the interval (−θ ξ /2, +θ ξ /2). We denote by δ + (ξ ) and δ − (ξ ) the greatest positive real numbers such that the strip is free of eigenvalues of the pencil A ξ (λ). Furthermore, we define Operator pencils A i (λ) for vertices: Let x (i) be a vertex of D, and let J i be the set of all indices j such that x (i) ∈ j . By our assumptions in Definition 2.1, there exist a is a cone with vertex at the origin, and˚ k = {x : x/|x| ∈ γ k } are the faces of this cone. Without loss of generality, we may assume that the Jacobian matrix κ (x) is equal to the identity matrix I at the point x (i) . We introduce spherical coordinates ρ = |x|, ω = x/|x| in K i , and define Remark 2.20 (Operator pencils for parabolic problems) Since we shall study parabolic PDEs, where the differential operator L(t, x, ∂ x ) additionally depends on the time t, we have to work with operator pencils A ξ (λ, t) and A i (λ, t) in this context. The philosophy is to fix t ∈ [0, T ] and define the pencils as above: We replace (2.5.2) by and work with δ (ξ )

Regularity results for parabolic PDEs
In this section we study the regularity of parabolic partial differential equations in weighted Sobolev spaces and (Hölder-)Besov spaces on domains of polyhedral type. We extend the works of Maz'ya and Rossmann [26] (where only elliptic problems were considered), Luong and Loi [24] (whose results were established on polyhedral cones only), and Schneider [28] (where regularity in Kondratiev spaces was studied, which are contained in our scale of weighted Sobolev spaces).
To provide Besov regularity for the parabolic problem with inhomogeneous boundary conditions we first shall prove two Sobolev regularity theorems (without and with time derivatives) for the inhomogeneous problem based on the homogeneous case from [28]. Then we give regularity assertions in V -spaces for homogeneous and inhomogeneous problems for domains of polyhedral type D. Finally, we deduce Besov and Hölder-Besov regularity by applying the embedding V l,2 β,δ (D) ∩ H s (D) → B r τ,τ (D) subject to some restrictions on the parameters.

Known auxiliary results
For our regularity assertions we rely on known results for elliptic equations when fixing the time parameter t. Therefore, we consider first the elliptic problem (2.5.1) (also with inhomogeneous boundary conditions for later use) and provide a lemma on the regularity of its solutions in domains of polyhedral type, which is a direct consequence of [26,Cor. 4 for all j = 1, . . . , n and k = 1, . . . , m. Suppose that the closed strip between the lines where C is a constant independent of u, f and g j,k .
Later on we rely also on the following lemma from [28, Lem. 5.2.1].

Lemma 3.2 (Continuity of bilinear form)
We recall [28, Thm. 5.2.2], which guarantees the existence (and uniqueness) of the weak solution of the homogeneous problem (P hom ) and provides Sobolev regularity without time derivatives for it.
where C is a constant independent of u and f .
The above theorem can be strengthened as follows to include Sobolev regularity results for the time derivatives as well, cf. [28,Thm. 5

.2.3].
Theorem 3.4 (Sobolev regularity with time derivatives for homogeneous problem) Let l ∈ N 0 and assume that the right hand side f of problem (P hom ) satisfies Then the weak solution u in the space H 1 ([0, T ],H m , H −m ) of problem (P hom ) has derivatives with respect to t up to order l satisfying where C is a constant independent of u and f .

Regularity results in Sobolev spaces for the inhomogeneous problem
In this section we generalize Theorems 3.3 and 3.4 to the inhomogeneous setting. Our main strategy for proving our regularity results in Sobolev spaces with inhomogeneous boundary conditions will be to reduce the inhomogeneous problem to a similar one with homogeneous boundary conditions. We briefly illustrate the concept for elliptic PDEs.
Consider the problem where g ∈ H m (D) is the inhomogeneous boundary condition and L an elliptic differential operator of the form (2.4.1) (independent of the time parameter t). Using the variational formulation, a weak solution of (3. for all ϕ ∈H m (D). This can be rewritten in terms of w := u − g. Then the corresponding (homogeneous) problem is to find w ∈H m (D) such that

2.4)
where C is a constant independent of f , g, u 0 , and u.

Remark 3.6
The restriction of the boundary data g at time t = 0 makes sense: By our assumptions g(0, ·) ∈ L 2 (D) is well-defined, since by (2.3.2) we have Proof First we show the uniqueness of the solution. Let u 1 , be two weak solutions of (P inhom ). Then are two weak solutions of the following homogeneous problem with the new right hand side F ∈ L 2 ([0, T ], H −m (D)). In this case we have for i = 1, 2 that

Subtraction yields
which proves the uniqueness of the solution of problem (P inhom ). Now we turn towards proving the existence of a solution. By the assumption a αβ = a βα for |α|, |β| ≤ m, L is formally a self-adjoint operator. Moreover, the Rellich-Kondrachov Theorem tells us that the embeddingH m (D) → L 2 (D) is compact (in particular, this holds for bounded Lipschitz domains and therefore also for bounded Lipschitz domains of polyhedral type).
From this we deduce that the operator L(0, x, ∂ x ) possesses a set {ψ k } ∞ k=1 consisting of all its eigenfunctions, which is not only an orthogonal basis ofH m (D) but also an orthonormal basis of L 2 (D). We refer to [15,Thm. 8.39] in this context (with X =H m , Y = L 2 ; the fact that the embeddingH m (D) → L 2 (D) is compact is crucial here). We now want to approximate w, the solution of problem (3.2.6). For each positive integer N consider the function is the solution of the ordinary differential equation system

2.9)
for k = 1, . . . , N . After multiplying both sides of (3.2.8) by C N k (t), taking the sum with respect to k from 1 to N , and integrating with respect to t from 0 to T (T > 0), we arrive at Applying Young's inequality, i.e., ab ≤ 1 2 εa 2 + b 2 ε , we can estimate term I by (3.2.12) For term I I we similarly obtain that (3.2.13) Applying Hölder's and again Young's inequality, we estimate for term I I I (3.2.15) Therefore, choosing 0 < 3ε < 2μ we get from (3.2.11)-(3.2.15) that (3.2.16) We now estimate term I V as follows: Put Then (  We now estimate the term η(0) using the definition of w N and the fact that {ψ k } ∞ k=1 is an orthonormal basis in L 2 (D) and obtain Then we obtain from (3.2.17) the estimate This inserted in (3.2.16) yields Therefore, Since v 1 |H m (D) ≤ 1, we see that thus, Integration over t together with (3.2.18) yields where in the second step we use the fact that and satisfies the initial-boundary conditions using w = u − g, (3.2.5) and the construction from (3.2.7), (3.2.9). This completes the proof.
To prove the generalization of Theorem 3.4 to the inhomogeneous setting we will need the following auxiliary lemma whose proof is based on [4,Lem. 4.3] with suitable modifications for our needs and which is related to the following problem: Problem (P 3 ) has the advantage that it is easier to handle since now homogeneous boundary conditions appear.
where C is a constant independent of f , ϕ 0 and w.

Remark 3.8
Later on we shall apply Lemma 3.7 to ϕ 0 := u 0 − g(0, ·) ∈H m (D) in order to relate problem (P 3 ) with our inhomogeneous problem (P inhom ). Thus, for the function u 0 we require u 0 ∈ H m (D) and for g we require that it is smooth enough such that This is satisfied if

Proof
Step 1: Let us consider first the case f ∈ L 2 (D T ). We use the same construction as in (3.2.7) but now for each positive integer N consider the function is the solution of the ordinary differential equation system , then taking the sum with respect to k from 1 to N , and afterwards integrating with respect to t from 0 to T , we arrive at Here we used that f , Using integration by parts with respect to t we obtain Furthermore, we use the estimate (3.2.18) replacing u 0 − g(0, ·) by ϕ 0 and g by the zero function and obtain

Combining (3.2.28) and (3.2.29) we obtain
where C is a constant independent of N , f , and ϕ 0 . Hence, we see that {w N } ∞ N =1 is a uniformly bounded sequence in the reflexive space which proves the lemma for X = L 2 (D T ). Step

.30) and Young's inequality twice we calculate
By the same argument as in Step 1 above, we conclude the existence of a convergent subse- Then if w 1 , w 2 are the solutions of (3.2.34) and (3.2.35), respectively, we deduce from Lemma 3.7 that w = w 1 + w 2 is the solution of (P 3 ) and the following inequality holds We now generalize Theorem 3.4 to inhomogeneous boundary conditions. We rely on the proof of the homogeneous case from [ We say that the l-th-order compatibility conditions are fulfilled if ∂ t k F ∈ L 2 ([0, T ], H (l−k)2m (D)) for k = 0, . . . , l, and additionally, it holds that ϕ ∈H (2 l+1)m (D), ϕ 0 , . . . , ϕ l−1 ∈H 2m (D), and ϕ l ∈H m (D).

Remark 3.11
Note that such compatibility conditions are not needed in Theorem 3.5 since this would simply correspond to require that u 0 − g(0, ·) = 0 on ∂ D (no further conditions in terms of derivatives w.r.t. time are needed). However, since g(0, ·) and u 0 by the assumptions belong to L 2 (D) (and are therefore not defined pointwise) such a condition is not necessary (does not make sense) in this context.
With these compatibility conditions we are now able to generalize Theorem 3.4 for the inhomogeneous problem (P inhom ) as follows.

2.37)
where C is a constant independent of f , g, u 0 , and u.
Proof We define the function w := u − g and consider the following equivalent problem to problem (P inhom ): In particular, problem (3.2.38) has the advantage that it is easier to handle compared to problem (P inhom ), since now homogeneous boundary conditions appear. In what follows, we will show the theorem by induction on l and additionally prove that the following equalities hold: In particular, F ∈ L 2 (D T ) follows from f ∈ L 2 (D T ) and the fact that the assumptions on g imply ∂ t g ∈ L 2 ([0, T ], L 2 (D)) and Lg ∈ L 2 ([0, T ], L 2 (D)).
Therefore, we obtain which inserted in (3.2.40) and using the embedding (3.2.20) gives This proves the claim for l = 0. Assuming that the assumptions hold for l − 1, i.e., we will prove them for l (l ≥ 1). We consider first the following problem: find a function   (H m (D)) ) where C is a constant independent of f , g, u 0 , and u. Hence, according to Theorem 3.5 the problem (3.

2.41) has a solution
where C is a constant independent of f , g, u 0 and u.
) and ϕ l |H m (D) can be estimated by the right hand side of (3.2.37). This finally completes the proof.

Regularity results in weighted Sobolev spaces
In this subsection we provide regularity assertions for the weak solution of problems (P hom ) and (P inhom ) in the spaces V l, p β,δ (D). Forerunners for the homogeneous case can be found in [24,Thm. 3.3], where the underlying domain is a polyhedral cone, and in [28,Thms. 5.2.9,5.2.12], where corresponding regularity results are proven on domains of polyhedral type in the scale of Kondratiev spaces (which is contained in the scale of V -spaces). We now want to generalize these results to the more general spaces V l, p β,δ (D) where the underlying domain is of polyhedral type for both, the homogeneous and the inhomogeneous setting.

Weighted Sobolev regularity for the homogeneous problem
We first investigate the regularity in weighted Sobolev spaces for the parabolic problem (P hom ) on domains of polyhedral type.

Moreover, suppose that the closed strip between the lines Reλ
for l = −1, 0, . . . , γ m . In particular, for the derivatives ∂ t l+1 u up to order γ m + 1 we have the a priori estimate where the constant is independent of u and f .
Moreover, we have where for fixed t the right hand side belongs to V 0 β,δ (D) and according to (2.1.4) Thus, an application of Lemma 3. 1 (withγ = m,δ k = 0, k = 1, . . . , d,β j = 0, j = 1, . . . , d , γ = 2m, β = with the a priori estimate . Now integration w.r.t. the parameter t together with (3.3.2) proves the claim for γ = 2m, i.e., Assume inductively that our assumption holds for γ − 1. This means, in particular, that we have the following a priori estimate We are going to show that the claim then holds for γ as well. If l = γ m by our assumptions where j = 1, . . . , d and k = 1, . . . , d. In particular, Theorem 3.4 provides us with the a priori estimate which shows the claim for l = γ m and arbitrary γ . Hence, the claim holds for l = γ m . We now proceed by backwards induction. Suppose now the result holds for γ and l = γ m , γ m −1, . . . , i where 0 ≤ i ≤ γ m . We show that it then also holds for i − 1. Differentiating (P hom ) i-times gives . From our initial assumptions on f we see where we used Lemma 2.5 and the fact that The differential operator ∂ t i−k L diminishes the space regularity by 2m since its coefficients a αβ are smooth functions. Thus, from (3.3.5) we see that hence, the right hand side of (3.  = 1, . . . , d andδ k = 0 for k = 1, . . . , d) yields Moreover, we have the a priori estimate where we used (3.3.5) in the last step. Integration w.r.t. the parameter t together with the inductive assumptions on ∂ t i+1 u and ∂ t k u (cf. (3.3.3)) gives the a priori estimate where in the last step we used the fact that (γ − 1) m ≤ γ m . This shows that the claim is true for i − 1 and completes the proof.

Remark 3.16 (Restrictions on the parameter δ)
We discuss the restrictions on the parameter δ = (δ 1 , . . . , δ d ) appearing in Theorem 3.14. For simplicity, let the domain of polyhedral type D be a polyhedron with straight edges and faces where θ k denotes the angle at the edge M k . Then we require for δ k , k = 1, . . . , d that We consider the case of the heat equation when m = 1 and δ (k)  [24,Sect. 4]. For general angles θ k we obtain the restrictions To guarantee that such δ k 's exist we require which shows that the greater the angle θ k , the smaller γ 1 (and thus γ ) has to be in order to satisfy the assumptions. In particular, for the extremal case θ k = 2π we require γ 1 = 0 (i.e., γ = 2 and 1 2 < δ k ≤ 1) whereas for θ k = π 4 we obtain γ 1 < 2 (i.e., γ < 4 with In this case it is always possible to find suitable δ k 's, since 1 − π θ k < 1 is always satisfied. If γ 1 = 1 (and i = 0) then max 2(γ 1 −i)− π θ k +1, −1 = max 3− π θ k , −1 . In this case we require that Thus, in this case we can find suitable δ k 's only for not too great angles, in particular, only for angles smaller than π 2 . Furthermore, if we have a non-convex angle θ k > π then only γ 1 = 0 is allowed, and hence, γ = 2 is required with max 1 − π θ k , −1 = 1 − π θ k < δ k ≤ 1. In this case it is possible to find suitable δ k 's but they cannot be negative since 1 − π θ k > 0. From the above discussion we see that the shape of the domain limits strongly the regularity of the solutions even for smooth data.
In general, we see that the largest angle of the edges determines the maximal (possible) range for the smoothness γ , which can be considered. For every vertex x j , j = 1, . . . , d , we can use that the truncation near the vertex is a polyhedral cone, and hence, for the edges at this vertex we can use the formula θ j 1 +· · ·+θ j l = (l −2)π. For fixed l the maximum of the angles is the smallest in the case of a polyhedron having the same angles, i.e., θ = θ j i = l−2 l π, i = 1, . . . , l. We see that the greater l is, the greater is also θ . Therefore, the 'best' case (when one wishes to consider large values of γ ) is the tetrahedron with angles θ = π 3 .
We now present our second result concerning weighted Sobolev regularity for the parabolic problem (P hom ) on domains of polyhedral type, where the right-hand side f now is arbitrarily smooth with respect to time.
Furthermore, suppose that the closed strip between the lines Re λ = m − 3/2 and Re λ = η − β j − 3/2 is free of eigenvalues of the pencil A j (λ) for j = 1, . . . , d and that In particular, for the derivatives ∂ t k u, k = 0, . . . , l, we have the a priori estimate where the constant is independent of u and f .
Proof We prove the theorem by induction. In particular, we first show the theorem for η = 2m and l = 0. Then from the induction assumption η = 2m, k = 0, . . . , l − 1 we make an induction step to η = 2m, k = l. Afterwards, with the induction assumption η − 1, l ∈ N 0 we prove the theorem for arbitrary η ≥ 2m and l ∈ N 0 .
Moreover, since where for fixed t the right hand side F(t) belongs to V 0 β,δ (D) (and due to the fact that β,δ (D) with the a priori estimate Now integration w.r.t. the parameter t and (3.3.9) prove the claim for η = 2m and l = 0, i.e., We now assume that the claim is true for η = 2m and k = 0, . . . , l −1. Then differentiating (P hom ) l-times gives β,δ (D) and we have the following a priori estimate Integration w.r.t. t, our inductive assumptions, and Theorem 3.4 give Assume now inductively that our assumption holds for η − 1 and all derivatives l ∈ N 0 . We are going to show first that the claim then holds for η and l = 0 as well. Looking at (3.3.10) we see that f (t) ∈ V η−2m β,δ (D) and by our inductive assumption and Lemma 2.5 we (D) and an application of Lemma 3.1 (withγ = m,β j = 0, j = 1, . . . , d ,δ k = 0, k = 1, . . . , d, γ = η, β = β, δ = δ) gives u(t) ∈ V η β,δ (D) with a priori estimate Integration w.r.t. the parameter t and our inductive assumptions show that the claim is true for η and l = 0, i.e., Suppose now that it is true for η and derivatives k = 0, . . . , l − 1. Differentiating (P hom ) ltimes again gives (3.3.11). From our initial assumptions on f we see that ∂ t l f (t) ∈ V η−2m β,δ (D) and from the inductive assumptions it follows that ∂ t l+1 u(t) ∈ V η−1 (D). Moreover, by the inductive assumptions ∂ t k u(t) ∈ V η β,δ (D) for k = 0, . . . , l − 1, and therefore, This shows that the right hand side in (3.3.11) Integration w.r.t. the parameter t together with our inductive assumptions gives which finishes the proof.

Remark 3.18
In Theorem 3.17 compared to Theorem 3.14 we require for δ k , k = 1, . . . , d only the parameter restrictions where η ≥ 2m. Focusing again on the heat equation with m = 1 and δ (k) Hence, we require Let now an angle θ k of D be non-convex, i.e., θ k > π, then only η = 2 is possible at best. Moreover, the restriction 1 − π θ k < δ k ≤ 1 leads to the same situation that we already discussed in Remark 3.16.
On the other hand, for smaller angles we obtain a better result compared to Theorem 3.14: If θ k = π 4 (or slightly below) then η < 6 is allowed (compared to γ 1 < 2 in Theorem 3.14 and hence, maximal regularity of u being 2(γ 1 + 1) = 4) and suitable δ k 's can be chosen.
Again we refer to the discussion in Remark 3.16 in this context. As was also mentioned there we deduce from the above considerations that the 'best' possible range for η is obtained for a tetrahedron with angles θ = π 3 : In this case η < 2 + 3 = 5, and thus, 2 ≤ η ≤ 4 with −1 < δ k ≤ 1 for η = 2, 3 and 0 < δ k ≤ 1 for η = 4.

Weighted Sobolev regularity for the inhomogeneous problem
We now generalize Theorem 3.14 to inhomogeneous boundary conditions. The proof is based on the homogeneous case subject to suitable modifications.
for l = −1, 0, . . . , γ m . In particular, for the derivatives ∂ t l+1 u up to order γ m + 1 we have the a priori estimate (3.3.12) where the constant is independent of f , g, u 0 , and u.

Remark 3.20
The existence of the solution . . , γ m + 1 in Theorem 3.19 follows from Theorems 3.5 and 3.12 with l = γ m + 1.
Proof We prove the theorem by induction. In particular, we first show the claim for the induction basis γ = 2m. Then we proceed with the induction step from γ − 1 to γ with l = γ m , and finally, we make a backwards induction from l = γ m , γ m − 1, . . . , i to l = i − 1 with arbitrary γ . We define the function w := u − g and consider again the following equivalent problem to problem (P inhom ): Let γ = 2m, then we have γ m = 0. Since by our assumptions f , ) and from the assumptions on g above we obtain that Moreover, we have where for fixed t the right hand side belongs to V 0 β,δ (D) (cf. (3.3.16) and the initial assumptions on f , g, u 0 ) and according to (2.1.4) β,δ (D) with the a priori estimate . Now integration w.r.t. the parameter t together with (3.3.16) yields Using that u = w + g from (3.3.18), the initial assumptions and the fact that the differential operator L diminishes the space regularity by 2m we calculate where we used (3.3.15) in the last step. This shows the claim for the induction basis γ = 2m. Assume inductively that our assumption holds for γ − 1. This means, in particular, that we have the following a priori estimate (3.3.19) We are going to show that the claim then holds for γ as well. Let l = γ m . Then by our assumptions on f , g using that ) and the fact that the differential operators ∂ t i L diminish the space regularity by 2m for i = 0, . . . , k (since the coefficients a αβ are smooth functions), where j = 1, . . . , d and k = 1, . . . , d. In particular, Theorem 3.12 provides us with the a priori estimate from which we obtain We first estimate term I and calculate From our initial assumptions on f and g we see that (D) and from the initial assumptions on g together with the inductive assumptions it follows for where we used Lemma 2.5 and the fact that (γ −1) m −(r −1) ≥ γ m −i +1 for r = 0, . . . , i −1.
Since the differential operator ∂ t i−k L diminishes the space regularity by 2m, from (3.3.24) we see that Hence, the right hand side of (3.
Moreover, we have the a priori estimate where we used (3.3.24) in the last step. From that we obtain Using the fact that and integrating (3.3.25) w.r.t. the parameter t together with the inductive assumptions on ∂ t i+1 u and ∂ t r u (cf. (3.3.19)) we obtain the a priori estimate where we used (γ − 1) m ≤ γ m . This shows that the claim is true for i − 1, which completes the proof.
We now generalize Theorem 3.17 to inhomogeneous boundary conditions. The proof is again based on the homogeneous case subject to suitable modifications.
and the (η − 2m + l + 1)-st order compatibility conditions from Definition 3.10 hold. Moreover, suppose that the closed strip between the lines Reλ = m −3/2 and Reλ = η −β j −3/2 is free of eigenvalues of the pencil A j (λ) for j = 1, . . . , d and that In particular, for the derivatives of u up to order l we have the a priori estimate

3.26)
where the constant is independent of f , g, u 0 , and u.
Proof We prove the theorem by induction. In particular, we first show the theorem for η = 2m and l = 0. Then from the induction assumption η = 2m, k = 0, . . . , l − 1 we make an induction step to η = 2m, k = l. Afterwards, with the induction assumption η − 1, l ∈ N 0 we prove the theorem for arbitrary η ≥ 2m and l ∈ N 0 . We define the function w := u − g and consider again the following equivalent problem to problem (P inhom ): Let η = 2m and l = 0. Since by our assumptions  Moreover, we have

3.29)
where for fixed t the right hand side belongs to V 0 β,δ (D) (cf. (3.3.28) and the initial assumptions on f , g) and according to (2.1.4) also to V −m 0,0 (D) if β j ≤ m, j = 1, . . . , d and δ k ≤ m, k = 1, . . . , d. Thus, an application of Lemma 3.1 (withγ = m,δ k = 0, k = 1, . . . , d, β,δ (D) with the a priori estimate Now integration w.r.t. the parameter t together with (3.3.28) yields Using that u = w+g from (3.3.30), the initial assumptions and the fact that the differential operator L diminishes the space regularity by 2m we calculate (D). This shows the claim for arbitrary η ≥ 2m and l ∈ N 0 , which finishes the proof.

Regularity results in Besov spaces
In this section we study regularity in Besov spaces for the parabolic problems (P hom ) and (P inhom ) with homogeneous and inhomogeneous boundary conditions, respectively, on domains of polyhedral type. We shall rely on our previous regularity results in weighted Sobolev spaces (cf. Theorems 3.14, 3.17, 3.19, and 3.21).

Besov regularity for the homogeneous problem
A combination of the regularity results from Theorems 3.4 and 3.14 together with the embedding from Theorem 2.10 yields the following result concerning Besov regularity of the homogeneous parabolic problem (P hom ). This completes the proof.
Alternatively, we combine Theorems 2.10, 3.4, and 3.17. This leads to the following regularity result in Besov spaces for the homogeneous problem (P hom ). where the restrictions on r in (3.4.4) immediately follow from the restrictions of Theorem 2.10. Moreover, η > β holds, since η ≥ 2m > m ≥ β. Furthermore, the a priori estimate

Hölder-Besov regularity for the homogeneous problem
We observe the fact that Theorems 3.14 and 3.17 not only provide regularity properties of the solution u of problem (P hom ) but also of its partial derivatives ∂ t k u. Following [28,Subsec. 5.4.3] in this subsection we use this fact in combination with the generalized Sobolev embedding (2.3.6) in order to obtain some mixed Hölder-Besov regularity results. where in the second step we require η ≤ 2m(γ m − k). Moreover, in the last step by Theorem 2.10 we get the additional restrictions η > β and r < min{η, 3(η − |δ + |), 3m} < min{2m(γ m − k), 3(2m(γ m − k) − |δ + |), 3m}.
In conclusion, we see that the generalization from Kondratiev to the V -spaces does not yield a better result in terms of the maximal Besov regularity. It would therefore be sufficient to work with the Kondratiev spaces in this context. However, a lot of the regularity results for parabolic PDEs on non-smooth domains are stated in terms of V -spaces, and therefore, it makes sense to use them directly. Our major contribution in this paper is that we are able to study the maximal Besov regularity of inhomogeneous parabolic problems as well, which has not been done so far.