Regularity in Sobolev and Besov spaces for parabolic problems on domains of polyhedral type

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale $B^r_{\tau,\tau}$, $\frac{1}{\tau}=\frac rd+\frac 1p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.


Introduction
This paper is concerned with regularity estimates of the solutions to evolution equations in nonsmooth domains of polyhedral type D ⊂ R 3 , cf. Definition 2.2. In particular, we study linear (ε = 0) and nonlinear (ε > 0) equations of the form with zero initial and Dirichlet boundary conditions, where m, M ∈ N, and L denotes a uniformly elliptic operator of order 2m with sufficiently smooth coefficients. Special attention is paid to the spatial regularity of the solutions to (1.1) in specific non-standard smoothness spaces, i.e., in the so-called adaptivity scale of Besov spaces B r τ,τ (D), Our investigations are motivated by fundamental questions arising in the context of the numerical treatment of equation (1.1). In particular, we aim at justifying the use of adaptive numerical methods for parabolic PDEs. Let us explain these relationships in more detail: 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. Although the basic idea is convincing, adaptive algorithms are hard to implement, so that beforehand a rigorous mathematical analysis to justify their use is highly desirable.
Given an adaptive algorithm based on a dictionary for the solution spaces of the PDE, the best one can expect is an optimal performance in the sense that it realizes the convergence rate of best N -term approximation schemes, which serves as a benchmark in this context. Given a dictionary Ψ = {ψ λ } λ∈Λ of functions in a Banach space X, the error of best N -term approximation is defined as i.e., as the name suggests we consider the best approximation by linear combinations of the basis functions consisting of at most N terms. In particular, [23,Thm. 11,p. 586] implies for τ < p, σ N u; L p (D) ≤ C N −s/d u|B s τ,τ (D) , Quite recently, it has turned out that the same interrelations also hold for the very important and widespread adaptive finite element schemes. In particular, [27,Thm. 2.2] gives direct estimates, where σ F E N denotes the counterpart to the quantity σ N (u; X), which corresponds to wavelet approximations. It can be seen that the achievable order of adaptive algorithms depends on the regularity of the target function in the specific scale of Besov spaces (1.2). On the other hand it is the regularity of the solution in the scale of Sobolev spaces, which encodes information on the convergence order for nonadaptive (uniform) methods. From this we can draw the following conclusion: adaptivity is justified, if the Besov regularity of the solution in the Besov scale (1.2) is higher than its Sobolev smoothness! For the case of elliptic partial differential equations, a lot of positive results in this direction are already established [13][14][15][16][17][18][19]30,31]. It is well-known that if the domain under consideration, the righthand side and the coefficients are sufficiently smooth, then the problem is completely regular [1], and there is no reason why the Besov smoothness should be higher than the Sobolev regularity. However, on general Lipschitz domains and in particular in polyhedral domains, the situation changes dramatically. On these domains, singularities at the boundary may occur that diminish the Sobolev regularity of the solution significantly [10,12,28,29,32]. However, the analysis in the above mentioned papers shows that these boundary singularities do not influence the Besov regularity too much, so that the use of adaptive algorithms for elliptic PDEs is completely justified! In this paper, we study similar questions for evolution equations of the form (1.1) and of associated semilinear versions. To the best of our knowledge, not so many results in this direction are available so far. For parabolic equations, first results for the special case of the heat equation have been reported in [2][3][4], but for a slightly different scale of Besov spaces. Our results show in the linear case ε = 0 that if the right-hand side as well as its time derivatives are contained in specific Kondratiev spaces, then, for every t ∈ [0, T ] the spatial Besov smoothness of the solution to (1.1) is always larger than 2m, provided that some technical conditions on the operator pencils are satisfied, see Theorems 5.1 and 5.4. The reader should observe that the results are independent of the shape of the polyhedral domain, and that the classical Sobolev smoothness is usually limited by m, see [35]. Therefore, for every t, the spatial Besov regularity is more than twice as high as the Sobolev smoothness, which of course justifies the use of (spatial) adaptive algorithms. Moreover, for smooth domains and right-hand sides in L 2 , the best one could expect would be smoothness order 2m in the classical Sobolev scale. So, the Besov smoothness on polyhedral type domains is at least as high as the Sobolev smoothness on smooth domains. Afterwards, we generalize this result to nonlinear parabolic equations of the form (1.1). We show that in a sufficiently small ball containing the solution of the corresponding linear equation, there exists a unique solution to (1.1) possessing the same Besov smoothness in the scale (1.2). The proof is performed by a technically quite involved application of the Banach fixed point theorem. The final result is stated in Theorem 5.6. The next natural step is to also study the regularity in time direction. For the linear parabolic problem (1.1) with ε = 0 we show that the mapping t → u(t, ·) is in fact a C l -map into the adaptivity scale of Besov spaces, precisely, see Theorem 5.8.
In conclusion, the results presented in this paper imply that for each t ∈ (0, T ) the spatial Besov regularity of the unknown solutions of the problems studied here is much higher than the Sobolev regularity, which justifies the use of spatial adaptive algorithms. This corresponds to the classical time-marching schemes such as the Rothe method. We refer e.g. to the monographs [34,42] for a detailed discussion. Of course, it would be tempting to employ adaptive strategies in the whole space-time cylinder. First results in this direction have been reported in [41]. To justify also these schemes, Besov regularity in the whole space-time cylinder has to be established. This case will be studied in a forthcoming paper. Throughout the paper we use the same notation as in [22], which for the convenience of the reader is recalled in Appendix A.1.

Sobolev and Kondratiev spaces
In this section we briefly collect the basics concerning weighted and unweighted Sobolev spaces needed later on. In particular, we put H m = W m 2 and denote byH m the closure of test functions in H m and its dual space by H −m . Moreover, C k,α , k ∈ N 0 , stands for the usual Hölder spaces with exponent α ∈ (0, 1]. The following generalized version of Sobolev's embedding theorem for Banach-space valued functions will be useful, cf. [40, Thm. 1.2.5]. Theorem 2.1 (Generalized Sobolev's embedding theorem) Let 1 < p < ∞, m ∈ N, I ⊂ R be some bounded interval, and X a Banach space. Then Here the Banach-valued Sobolev spaces are endowed with the norm whereas for the Hölder spaces we use where u|C k (I, X) = k j=0 max t∈I u (j) (t)|X and |u (k) | C α (I,X) = sup s,t∈I, We collect some notation for specific Banach-space valued Lebesgue and Sobolev spaces, which will be used when studying the regularity of solutions of parabolic PDEs. Let

Kondratiev spaces
In the sequel we work to a great extent with weighted Sobolev spaces, the so-called Kondratiev spaces K m p,a (O), defined as the collection of all u ∈ D (O), which have m generalized derivatives satisfying where a ∈ R, 1 < p < ∞, m ∈ N 0 , α ∈ N n 0 , and the weight function : D → [0, 1] is the smooth distance to the singular set of O, i.e., is a smooth function and in the vicinity of the singular set S it is equivalent to the distance to that set. Clearly, if O is a polygon in R 2 or a polyhedral domain in R 3 , then the singular set S consists of the vertices of the polygon or the vertices and edges of the polyhedra, respectively. It follows directly from (2.2) that the scale of Kondratiev spaces is monotone in m and a, i.e., if m < m and a < a. Moreover, generalizing the above concept to functions depending on the time t ∈ [0, T ], we define Kondratiev type spaces, denoted by L q ((0, T ), K m p,a (O)), which contain all functions u(x, t) such that with 0 < q ≤ ∞ and parameters a, p, m as above.
Kondratiev spaces on domains of polyhedral type For our analysis we make use of several properties of Kondratiev spaces that have been proved in [20]. Therefore, in our later considerations, we will mainly be interested in (a) The boundary ∂D consists of smooth (of class C ∞ ) open two-dimensional manifolds Γ j (the faces of D), j = 1, . . . , n, smooth curves M k (the edges), k = 1, . . . , l, and vertices x (1) , . . . , x (l ) .
which in polar coordinates can be described as Figure 1: Polyhedron where the opening angle θ of the 2-dimensional wedge K satisfies 0 < θ ≤ 2π.
(c) For every vertex x (i) there exists a neighbourhood U i and a diffeomorphism κ i mapping D ∩ U i onto K i ∩ B 1 (0), where K i is a polyhdral cone with edges and vertex at the origin.

Remark 2.3 (i)
In the literature many different types of polyhedral domains are considered. A more general version which coincides with the above definition when d = 3 is discussed in [20]. Further variants of polyhedral domains can be found in Babuška, Guo [7], Bacuta, Mazzucato, Nistor, Zikatanov [8] and Mazzucato, Nistor [37].
(ii) Let us point out that 'smooth' domains without edges and/or vertices are admissible in Defintion 2.2. We discuss this further in Section 3.2.
Some properties of Kondratiev spaces Concerning pointwise multiplication the following results are proven in [20]. holds for all u, v ∈ K m a,p (D).
Our main tool when investigating the Besov regularity of solutions to the PDEs will be the following embedding result between Kondratiev and Besov spaces, which is an extension of [30,Thm. 1]. A proof may be found in [40,Thm. 1.4.12].

Parabolic PDEs and operator pencils
In the sequel we deal with two different parabolic settings, Problems I and II, which are of general order and defined on domains of polyhedral type according to Definition 2.2. In particular, Problem II is the nonlinear version of Problem I and we investigate the spatial Besov regularity of the solutions of these two problems and to some extent also the Hölder regularity with respect to the time variable of Problem I.

The fundamental parabolic problems
Let D denote some domain of polyhedral type in R d according to Definition 2.2 with faces Γ j , j = 1, . . . , n. For 0 < T < ∞ put D T = (0, T ] × D and Γ j,T = [0, T ] × Γ j .
We will investigate the Besov regularity of the following linear parabolic problem.
Problem I (Linear parabolic problem in divergence form) Let m ∈ N. We consider the following first initial-boundary value problem . . , m, j = 1, . . . , n, Here f is a function given on D T , ν denotes the exterior normal to Γ j,T , and the partial differential operator L is given by where a αβ are bounded real-valued functions from C ∞ (D T ) with a αβ = (−1) |α|+|β| 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. Moreover, for simplicity we set Remark 3.1 (Assumptions on the time-dependent bilinear form) When dealing with parabolic problems it will be reasonable to suppose that B(t, ·, ·) satisfies for all u ∈H m (D) and a.e. t ∈ [0, T ]. We refer to [40,Rem. 2.3.5] for a detailed discussion.
It is our intention to also study nonlinear versions of Problem I. Therefore, we modify (3.1) as follows.
Problem II (Nonlinear parabolic problem in divergence form) Let m, M ∈ N and ε > 0. We consider the following nonlinear parabolic problem The assumptions on f and the operator L are as in Problem I. When we establish Besov regularity results for Problem II we interpret (3.6) as a fixed point problem and show that the regularity estimates for Problem I carry over to Problem II, provided that ε is sufficiently small.

Operator pencils
In order to correctly state the global regularity results in Kondratiev spaces for Problems I and II, we need to work with operator pencils generated by the corresponding elliptic problems in the polyhedral type domain D ⊂ R 3 . We briefly recall the basic facts needed in the sequel. For further information on this subject we refer to [33] The singular set S of D then is given by the boundary points M 1 ∪ . . . ∪ M l ∪ {x (1) , . . . , x (l ) }. We do not exclude the cases l = 0 (corner domain) and l = 0 (edge domain). In the last case, the set S consists only of smooth non-intersecting edges. Figure 2 gives examples of polyhedral domains without edges or corners, respectively. The elliptic boundary value problem (3.7) on D generates two types of operator pencils for the edges M k and for the vertices x (i) of the domain, respectively.

1) Operator pencil A ξ (λ) for edge points:
The pencils A ξ (λ) for edge points ξ ∈ M k are defined as follows: According to Definition 2.2 there exists a neighborhood U ξ of ξ and a diffeomor- Let Γ k ± be the faces adjacent to M k . Then by D ξ we denote the dihedron which is bounded by the half-planesΓ k ± tangent to Γ k ± at ξ and the edge M ξ =Γ k + ∩Γ k − . Furthermore, let r, ϕ be polar coordinates in the plane perpendicular to M ξ such thatΓ We define the operator pencil A ξ (λ) as follows: denotes the main part of the differential operator L(x, D x ) with coefficients frozen at ξ. This way we obtain in (3.8) a boundary value problem for the function U on the 1-dimensional subdomain I ξ with the complex parameter λ. Obviously, A ξ (λ) is a polynomial of degree 2m in λ.
The operator A ξ (λ) realizes a continuous mapping for every λ ∈ C. Furthermore, A ξ (λ) is an isomorphism for all λ ∈ C with the possible exception of a denumerable set of isolated points, the spectrum of A ξ (λ), which consists of its eigenvalues with finite algebraic multiplicities: Here a complex number λ 0 is called an eigenvalue of the pencil ± the largest positive real numbers such that the strip is free of eigenvalues of the pencil A ξ (λ). Furthermore, we put For example, concerning the Dirichlet problem for the Poisson equation on a domain D ⊂ R 3 of polyhedral type, the eigenvalues of the pencil A ξ (λ) are given by where θ ξ is the inner angle at the edge point ξ, cf. [40,Ex. 2.5.2]. Therefore, the first positiv eigenvalue is λ 1 = π θ ξ and we obtain δ ± = π θ ξ , cf. [40, Ex. 2.5.1].
2) Operator pencil A i (λ) for corner points: is a polyhedral cone with edges and vertex at the origin. W.l.o.g. we may assume that the Jacobian matrix κ i (x) is equal to the identity matrix at the point x (i) . We introduce spherical coordinates ρ = |x|, ω = x |x| in K i and define the operator pencil Furthermore, it is known that A i (λ) is an isomorphism for all λ ∈ C with the possible exception of a denumerable set of isolated points. The mentioned enumerable set consists of eigenvalues with finite algebraic multiplicities.
Moreover, the eigenvalues of A i (λ) are situated, except for finitely many, outside a double sector |Reλ| < ε|Imλ| containing the imaginary axis, cf. Remark 3.2 (Operator pencils for parabolic problems) Since we study parabolic PDEs, where the differential operator L(t, x, D 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 (3.8) by and work with δ ± (t) in (3.9) and (3.10), respectively. Moreover, we put δ

Regularity results in Sobolev and Kondratiev spaces
This section presents regularity results for Problems I and II in Sobolev and Kondratiev spaces. They will form the basis for obtaining regularity results in Besov spaces later on via suitable embeddings. The results in Sobolev and Kondratiev spaces for Problems I and II on domains of polyhedral type D ⊂ R d are essentially new and not published elsewhere so far: In [22] we restricted our investigations to polyhedral cones K ⊂ R 3 relying on the results from [35]. However, the extension of the regularity results for Problem I to polyhedral type domains follows from very similar arguments as in [22], which is why we merely state the results in Sections 4.1 and 4.2 and give references for the proofs wherever necessary. In contrast to this the regularity results for the nonlinear Problem II require some careful adaptations and are carried out in detail in Section 4.3.

Regularity results in Sobolev spaces for Problem I
In this subsection we are concerned with the Sobolev regularity of the weak solution of Problem I. We start with the following lemma, whose proof is similar to [5,Lem. 4.1].
Using the spectral method the following regularity result now follows.
where C is a constant independent of f and u.  and Then the weak solution u in the space H m,1 * (D T ) of Problem I in fact belongs to H m,l+1 * (D T ), i.e., has derivatives with respect to t up to order l satisfying where C is a constant independent of u and f .

Regularity results in Kondratiev spaces for Problem I
Concerning weighted Sobolev regularity of Problem I first fundamental results on polyhedral cones K ⊂ R 3 can be found in [35,Thms. 3.3,3.4]. In [22] we extended and generalized these results, which we now wish to transfer to our setting of polyhedral type domains D ⊂ R 3 . For our regularity assertions we rely on known results for elliptic equations. Therefore, we consider first the following Dirichlet problem for elliptic equations where D ⊂ R 3 is a domain of polyhedral type according to Definition 2.2 with faces Γ j . Moreover, we assume that x is a uniformly elliptic differential operator of order 2m with smooth coefficients A α . We need the following technical assumptions in order to state the Kondratiev regularity of (4.4).
Assumption 4.5 (Assumptions on operator pencils) Consider the operator pencils A i (λ, t), i = 1, . . . , l for the vertices and A ξ (λ, t) with ξ ∈ M k , k = 1, . . . , l for the edges of the polyhedral type domain D ⊂ R 3 introduced in Section 3.2. For the elliptic problem (4.4) we may drop t from the notation of the pencils, otherwise (for our parabolic problems) we assume t ∈ [0, T ] is fixed. Let K γ p,b (D) and K γ p,b (D) be two Kondratiev spaces, where the singularity set S of D is given by (1) , . . . , x (l ) } and weight parameters b, b ∈ R. Then we assume that the closed strip between the lines ± are defined in (3.10) (replaced by (3.12) for parabolic problems).
Suppose that K γ 2,a (D) and K γ 2,a (D) satisfy Assumption 4.5. Then u ∈ K γ 2,a +2m (D) and where C is a constant independent of u and F .  where the latter embedding follows from the corresponding duality assertion, i.e., we have K m 2,m (D) → K 0 2,0 (D) since m ≥ 0. In this case the solution u of Problem I satisfies . Note that all our arguments with u(t) and f (t), respectively, hold for a.e. t ∈ [0, T ]. However, since lower order time derivatives are continuous w.r.t. suitable spaces (but not necessarily the highest one, cf. the proof of Thm. 5.8), we will suppress this distinction in the sequel.
Using similar arguments as in [35,Thm. 3.3] we are now able to show the following regularity result in Kondratiev spaces. The proof follows along the same lines as [22,Thm. 3.6].
Furthermore, let Assumption 4.5 hold for weight parameters b = a+2m(γ m −i), where i = 0, . . . , γ m , and b = −m. Then for the weak solution u ∈ H m,γm+2 * (D T ) of Problem I we have 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 . The regularity results obtained in Theorem 4.9 only hold under certain restrictions on the parameter a we are able to choose. In particular, we cannot choose γ m > 0 if we have a non-convex polyhedral type domains D, since there is no suitable a satisfying all of our requirements in this case. In order to treat non-convex domains as well, we impose stronger assumptions on the right-hand side f , requiring that it is arbitrarily smooth w.r.t. the time. This additional assumption allows for a larger range of a. However, as a drawback, these results are hard to apply to nonlinear equations since the right-hand sides are not taken from a Banach or quasi-Banach space. The proof of the following theorem is similar to [22,Thm. 3.9] adapted to our setting.
Furthermore, let Assumption 4.5 hold for weight parameters b = a and b = −m. Then for the weak In particular, for the derivative ∂ t l u we have the a priori estimate where the constant is independent of u and f . − < a + m < δ (k) + independent of the regularity parameter η which can be arbitrarily high. In particular, for the heat equation on a domain of polyhedral type D (which for simplicity we assume to be a polyhedron with straight edges and faces where θ k denotes the angle at the edge M k ), we have δ (k) ± = π θ k , which leads to the restriction −1 ≤ a < min 1, π θ k − 1 . Therefore, even in the extremal case when θ k = 2π we can still take −1 ≤ a < − 1 2 (resulting in u ∈ L 2 ([0, T ], K η a+2 (D)) being locally integrable since a + 2 > 0). Then choosing η arbitrary high, we also cover non-convex polyhedral type domains with our results from Theorem 4.11.

Regularity results in Sobolev and Kondratiev spaces for Problem II
In this subsection we show that the regularity estimates in Kondratiev and Sobolev spaces as stated in Theorems 4.9 and 4.3, respectively, carry over to Problem II, provided that ε is sufficiently small. In order to do this we interpret Problem II as a fixed point problem in the following way. Let D and S be Banach spaces ( D and S will be specified in the theorem below) and letL −1 : D → S be the linear operator defined viaL where N : S → D is a nonlinear operator. If we can show that N maps S into D, then a solution of Problem II is a fixed point of the problem Our aim is to apply Banach's fixed point theorem, which will also guarantee uniqueness of the solution, if we can show that T := (L −1 • N ) : S 0 → S 0 is a contraction mapping, i.e., there exists some q ∈ [0, 1) such that where the corresponding subset S 0 ⊂ S is a small closed ball with centerL −1 f (the solution of the corresponding linear problem) and suitably chosen radius R > 0.
Our main result is stated in the theorem below.  and consider the solution space S := S 1 ∩ S 2 . Suppose that f ∈ D and put η := f | D and r 0 > 1. Moreover, we choose ε > 0 so small that where c > 0 denotes the constant in (4.28) resulting from our estimates below. Then there exists a unique solution u ∈ S 0 ⊂ S of Problem II, where S 0 denotes a small ball aroundL −1 f (the solution of the corresponding linear problem) with radius R = (r 0 − 1)η L −1 .
P r o o f : Let u be the solution of the linear problemLu = f . From Theorems 4.9 and 4.3 we know thatL −1 : D → S is a bounded operator. If u M ∈ D (this will immediatelly follow from our calculations in Step 1 as explained in Step 2 below), the nonlinear part N satisfies the desired mapping properties, i.e., N u = f − εu M ∈ D and we can apply Theorem 4.9 now with right-hand side N u.
Step 1: Since one sees thatL −1 • N is a contraction if, and only, if We analyze the resulting condition with the help of the formula This together with Theorem 4.9 gives Concerning the derivatives, we use Leibniz's formula twice and we see that (4.13) In order to estimate the terms ∂ t r u j and ∂ t k−l−r v M −1−j we apply Faà di Bruno's formula where the sum runs over all r-tuples of nonnegative integers (k 1 , . . . , k r ) satisfying 1 · k 1 + 2 · k 2 + . . . + r · k r = r. In particular, from (4.15) we see that k r ≤ 1, where r = 1, . . . , k. Therefore, the highest derivative ∂ t r u appears at most once. We apply the formula to g = u and f (x) = x j and make use of the embeddings ( For k = γ m we use Theorem 2.4(ii). (Note that in Theorem 2.4(ii) we require that 'a − 1 ≥ d p − 2' with d = 3 for the parameter. In our situation below a − 1 has to be replaced by a, which leads to our restriction a ≥ d 2 − 2 = − 1 2 .) Similar as above we obtain ∂ t r u j |K 0 2,a (D) ≤ c r,j k 1 +...+kr ≤j, 1·k 1 +2·k 2 +...+r·kr =r k 1 +...+kr ≤j, 1·k 1 +2·k 2 +...+r·kr =r ..+kr ≤j, 1·k 1 +2·k 2 +...+r·kr =r u|K 2mγm 2,a+2mγm (D) Note that we require γ m ≥ 1 in the last step. We proceed similarly for ∂ t k−l−r v M −1−j . Now (4.13) together with (4.16) and (4.17) inserted in (4.12) together with Theorem 2.4 give .

(4.20)
We give some explanations concerning the estimate above. In (4.18) the term with k = γ m requires some special care since we have to apply Theorem 2.4 (ii). In this case we calculate The lower order derivatives in the last line cause no problems since we can (again) apply Theorem 2.4(i) as before. The term u j v M −1−j |K 2 2,a+2 (D) can now be further estimated with the help of Theorem 2.4(i). For the term γm r=0 (∂ t r u j )(∂ t γm−r v M −1−j )|K 0 2,a (D) we again use Theorem 2.4(ii), then proceed as in (4.17) and see that the resulting estimate yields (4.18). Moreover, in (4.19) we used the fact that in the step before we have two sums with κ 1 + . . . + κ r ≤ j and κ 1 + . . . + κ k−l−r ≤ M − 1 − j, i.e., we have k − l different κ i 's which leads to at most k different κ i 's if l = 0. We allow all combinations of κ i 's and redefine the κ i 's in the second sum leading to κ 1 , . . . , κ k with κ 1 + . . . + κ k ≤ M − 1 and replace the old conditions κ 1 + 2κ 2 + rκ r ≤ r and κ 1 + 2κ 2 + (k − l − r)κ k−l−r ≤ k − l − r by the weaker ones κ 1 + . . . + κ k ≤ k and κ k ≤ 1. This causes no problems since the other terms appearing in this step do not depend on κ i apart from the product term. There, the fact that some of the old κ i 's from both sums might coincide is reflected in the new exponent 4κ i . From Theorem 2.1 we conclude that where we put η := f |D in the last line, c 0 denotes the constant resulting from (4.16) and (4.20) and c 2 = c 0 c 1 with c 1 being the constant from the estimates in Theorem 4.9. We now turn our attention towards the second term L −1 u M − v M |D 2 in (4.12) and calculate where we used Leibniz's formula twice as in (4.13) in the second but last line. Again Faà di Bruno's formula, cf. (4.14), is applied in order to estimate the derivatives in (4.22 (we remark that this is exactly the point where our assumption m ≥ 2 comes into play). With this we obtain Similar for ∂ t k−l−r v M −1−j . As before, from (4.15) we observe k r ≤ 1, therefore the highest derivative u (r) appears at most once. Note that since H m (D) is a multiplication algebra for m > d 2 , we get (4.24) with L 2 (D) replaced by H m (D) as well. Now (4.23) and (4.24) inserted in (4.22) gives where we put η := f |D and c 3 denotes the constant arising from our estimates (4.25) and (4.26) above. Now (4.12) together with (4.21) and (4.27) yields where c = c 2 + c 3 . ForL −1 • N to be a contraction, we therefore require cf. (4.11). In case of max(R + L −1 η, 1) = 1 this leads to On the other hand, if max(R + L −1 η, 1) = R + L −1 η, we choose R = (r 0 − 1)η L −1 , which gives rise to the condition . (4.30) Step 2: The calculations in Step 1 show that u M ∈ D: The fact that u M ∈ D 1 ∩ D 2 follows from the estimate (4.28). In particular, taking v = 0 in (4.28) we get an estimate from above for u M | D . The upper bound depends on u|S and several constants which depend on u but are finite whenever we have u ∈ S, see also (4.20) and (4.25). The dependence on R in (4.28) comes from the fact that we choose u ∈ B R (L −1 f ) in S there. However, the same argument can also be applied to an arbitrary u ∈ S; this would result in a different constantc. In order to have u M ∈ D, we still need to show that Tr ∂ t k u M = 0, k = 0, . . . , γ m . This follows from the same arguments as in [22,Thm. 4.10]: ) we see that the trace operator Tr (∂ t k u) := (∂ t k u) (0, ·) is well defined for k = 0, . . . , γ m + 1. Using the initial assumption u(0, ·) = 0 in Problem II, by density arguments (C ∞ (D T ) is dense in S) and induction we deduce that (∂ t k u)(0, ·) = 0 for all k = 0, . . . , γ m + 1. Moreover, since by Theorem 2.1 we see that the trace operator Tr ∂ t k u M := ∂ t k u M (0, ·) is well defined for k = 0, . . . , γ m . By (4.24) below the term ∂ t k u M (0, ·)|L 2 (D) is estimated from above by powers of (∂ t l u) (0, ·)|H m (D) , l = 0, . . . , k. Since all these terms are equal to zero, this shows that u M ∈ D.
Step 3: The next step is to show that ( we only need to apply the above estimate (4.28) with v = 0. This gives which, in case that max(R + L −1 η, 1) = 1, leads to whereas for max(R + L −1 η, 1) = R + L −1 η we get We see that condition (4.31) implies (4.29). Furthermore, since , also condition (4.32) implies (4.30). Thus, by applying Banach's fixed point theorem in a sufficiently small ball around the solution of the corresponding linear problem, we obtain a unique solution of Problem II.
Remark 4.14 The restriction m ≥ 2 in Theorem 4.13 comes from the fact that we require s 2 = m > d 2 = 3 2 in (4.23). This assumption can probably be weakened, since we expect the solution to satisfy u ∈ L 2 ([0, T ], H s (D)) for all s < 3 2 , see also Remark 5.3 and the explanations given there. Moreover, the restriction a ≥ − 1 2 in Theorem 4.13 comes from Theorem 2.4(ii) that we applied. Together with the restriction a ∈ [−m, m] we are looking for a ∈ [− 1 2 , m] if the domain is a corner domain, e.g. a smooth cone K ⊂ R 3 (subject to some truncation). For polyhedral cones with edges M k , k = 1, . . . , l, we furthermore require −δ

Regularity results in Besov spaces
With all preliminary work, in this section we finally come to the presentation of the regularity results in Besov spaces for Problems I and II. For this purpose, we rely on the results from Section 4 on regularity in Sobolev and Kondratiev spaces for the respective problems and combine these with the embeddings of Kondratiev spaces into Besov spaces. It turns out that in all cases studied the Besov regularity is higher than the Sobolev regularity. This indicates that adaptivity pays off when solving these problems numerically. The Sobolev regularity we are working with (e.g. see Theorem 4.2 for Problem I) canonically comes out from the variational formulation of the problem, i.e., we have spatial Sobolev regularity m if the corresponding differential operator is of order 2m. We give an outlook on how our results could be improved by using regularity results in fractional Sobolev spaces instead. It is planned to do further investigations in this direction in the future. Moreover, we discuss the role of the weight parameter a appearing in our Kondratiev spaces to some extent.

Besov regularity of Problem I
A combination of Theorem 4.9 (Kondratiev regularity A) and the embedding in Theorem 2.5 yields the following Besov regularity of Problem I. Let γ ∈ N with γ ≥ 2m and put γ m := γ−1 2m . Furthermore, let a ∈ R with a ∈ [−m, m]. Assume that the right-hand side f of Problem I satisfies where 1 2 < 1 τ < α d + 1 2 and δ denotes the dimension of the singular set of D. In particular, for any α satisfying (5.1) and τ as above, we have the a priori estimate P r o o f : According to Theorem 4.9 by our assumptions we know u ∈ L 2 ([0, T ], K 2m(γm+1) 2,a+2m(γm+1) (D)). Together with Theorem 2.5 (choosing k = 0) we obtain where in the third step we use the fact that 2m(γ m + 1) ≥ 2m γ 2m − 1 + 1 = γ and choose α ≤ γ. Moreover, the condition on a from Theorem 2.5 yields m =min(m, a + 2m(γ m + 1)) > δ 3 α. Therefore, the upper bound for α is α < min γ, 3 δ m . Concerning the restriction on τ , Theorem 2.5 with τ 0 = 2 gives 1 2 < 1 τ < 1 τ * = α 3 + 1 2 . This completes the proof. 2,a+2m(γm+1) (D) we see that we require hence, if a < 0 the derivatives of the solution u might be unbounded near the boundary of D. From this it follows that the range −m < a < 0 is the most interesting for our considerations. where a = a + 2m(γ m + 1) ≥ a + 2m and again 1 2 < 1 τ < α 3 + 1 2 but the restriction on α now reads as For general Lipschitz domains D ⊂ R 3 we expect that the solution of Problem I (for m = 1) is contained in H s (D) for all s < 3 2 (as in the elliptic case, cf. [32]). This would lead to α < 9 2 when δ = 1. For convex domains it probably even holds that s = 2 (for the heat equation this was already proven in [45]). First results in this direction can be found in [21].

Besov regularity of Problem II
Concerning the Besov regularity of Problem II, we proceed in the same way as before for Problem I: Combining Theorem 4.13 (Nonlinear Sobolev and Kondratiev regularity) with the embeddings from Theorem 2.5 we derive the following result.
Theorem 5.6 (Nonlinear Besov regularity) Let the assumptions of Theorems 4.13 and 4.9 be satisfied. In particular, as in Theorem 4.13 for η := f | D and r 0 > 1, we choose ε > 0 so small that

5)
and Then there exists a solution u of Problem II, which satisfies u ∈ B 0 ⊂ B, for all 0 < α < min 3 δ m, γ , where δ denotes the dimension of the singular set of D, 1 2 < 1 τ < α 3 + 1 2 , and B 0 is a small ball aroundL −1 f (the solution of the corresponding linear problem) with radius R = CC(r 0 − 1)η L −1 .L To be more precise, Theorem 4.13 establishes the existence of a fixed point u in This together with the embedding results for Besov spaces from Theorem 2.5 (choosing k = 0) completes the proof, in particular, we calculate for the solution (cf. the proof of Theorem 5.1) Furthermore, it can be seen from (5.7) that new constants C andC appear when considering the radius R around the linear solution where the problem can be solved compared to Theorem 4.13.
Remark 5.7 A few words concerning the parameters appearing in Theorem 5.6 (and also Theorem 4.13) seem to be in order. Usually, the operator norm L −1 as well as ε are fixed; but we can change η and r 0 according to our needs. From this we deduce that by choosing η small enough the condition (5.6) can always be satisfied. Moreover, it is easy to see that the smaller the nonlinear perturbation ε > 0 is, the larger we can choose the radius R of the ball B 0 where the solution of Problem II is unique.

Hölder-Besov regularity of Problem I
So far we have not exploited the fact that Theorem 4.9 (Kondratiev regularity A) not only provides regularity properties of the solution u of Problem I but also of its partial derivatives ∂ t k u. We use this fact in combination with Theorem 2.1 in order to obtain some mixed Hölder-Besov regularity results on the whole space-time cylinder D T . For parabolic SPDEs, results in this direction have been obtained in [9]. However, for SPDEs, the time regularity is limited in nature. This is caused by the non-smooth character of the driving processes. Typically, Hölder regularity C 0,β can be obtained, but not more. In contrast to this, it is well-known that deterministic parabolic PDEs are smoothing in time. Therefore, in the deterministic case considered here, higher regularity results in time can be obtained compared to the probabilistic setting.
(ii) From the proof of Theorem 5.8 above it can be seen that the solution satisfies u ∈ C k, 1 2 ([0, T ], K 2m(γm−k) 2,a+2m(γm−k) (D)), implying that for high regularity in time, which is displayed by the parameter k, we have less spatial regularity in terms of 2m(γ m − k). suitable building blocks, cf. [43,44] and the references therein. Under certain restrictions on the parameters these different approaches might even coincide. Throughout this paper we rely on the characterization of Besov spaces via wavelet decompositions and refer in this context to [11,38]. Let us briefly recall the concept: Wavelets are specific orthonormal bases for L 2 (R) that are obtained by dilating, translating and scaling one fixed function, the so-called mother wavelet ψ. The mother wavelet is usually constructed by means of a so-called multiresolution analysis, that is, a sequence {V j } j∈Z of shift-invariant, closed subspaces of L 2 (R) whose union is dense in L 2 while their intersection is trivial. Moreover, all the spaces are related via dyadic dilation, and the space V 0 is spanned by the translates of one fixed function φ, called the generator. In her fundamental work [24,25] I. Daubechies has shown that there exist families of compactly supported wavelets. By taking tensor products, a compactly supported orthonormal basis for L 2 (R d ) can be constructed. Let φ be a father wavelet of tensor product type on R d and let Ψ = {ψ i : i = 1, . . . , 2 d − 1} be the set containing the corresponding multivariate mother wavelets such that, for a given r ∈ N and some N > 0 the following localization, smoothness and vanishing moment conditions hold. For all ψ ∈ Ψ , We refer again to [24,25] for a detailed discussion. The set of all dyadic cubes in R d with measure at most 1 is denoted by It follows that φ k , ψ I : k ∈ Z d , I ∈ D + , ψ ∈ Ψ is an orthonormal basis in L 2 (R d ). Denote by Q(I) some dyadic cube (of minimal size) such that supp ψ I ⊂ Q(I) for every ψ ∈ Ψ . Then, we clearly have Q(I) = 2 −j k + 2 −j Q for some dyadic cube Q. Put Λ = D + × Ψ . Then, every function f ∈ L 2 (R d ) can be written as It will be convenient to include φ into the set Ψ . We use the notation φ I := 0 for |I| < 1, φ I = φ(·−k) for I = k + [0, 1] d , and can simply write We describe Besov spaces on R d by decay properties of the wavelet coefficients, if the parameters fulfill certain conditions.