Hölder-estimates for non-autonomous parabolic problems with rough data

In this paper we establish Holder estimates for solutions to nonautonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding 
 elliptic operators are of divergence type, the coefficient matrix of 
 which depends only measurably on time. These results are in the 
 tradition of the classical book of Ladyshenskaya et al. [40], 
 which also serves as the starting point for our investigations.

1. Introduction. Parabolic equations are one of the most common features when modelling phenomena in science and engineering, see [2] and [12]. One of the main problems, however, is that the input to the equations is very often (highly) nonsmooth: the corresponding domains are not smooth (often they are not even strong Lipschitz domains), the coefficient functions are definitely discontinuous, and the boundary conditions are mixed: on one part D of the boundary Dirichlet conditions are imposed, while on the complement N Neumann-or Robin conditions hold. In the meantime these phenomena are also well investigated -as long as the coefficients do not depend explicitly on time, see [36], [33], [35] and [6]. In this paper we intend to investigate non-autonomous equations which incorporate all the phenomena described above with the central aim being Hölder estimates. This is also classical ever since the monography [40], as long as mixed boundary conditions are not considered.
Unfortunately, those investigations contain -in their generality -some peculiarities which make it not easy to apply them to problems originating from the applications: First, the Hölder spaces under consideration, see [40, pg. 7], are not the classical ones -the oscillation of the function is only measured over the connected components of the intersection of the domain with suitable balls (what is indeed adequate in case of general Dirichlet boundary data). Secondly, the estimates affect distributional right hand sides which are represented as the (spatial) Let us introduce the basic assumption on Ω and D which will define our geometrical framework and which is of fundamental importance in the sequel, cf.  I) If x ∈ (∂Ω \ N ), there is a domain U x =: U with x ∈ U, such that U ∩ N = ∅ and U ∩ Ω has only finitely many connected components V 1 , . . . , V k , where x is a limit point of each V j . Moreover, for every j ∈ {1, . . . , k}, there exists a number τ j > 0, an open neighbourhood U j of x satisfying V j ⊆ U j ⊆ U, and a bi-Lipschitz mapping φ j , defined on an open neighbourhood of U j into R n , such that φ j (x) = 0, φ j (U j ) = τ jK , φ j (V j ) = τ j K and φ j (∂V j ∩ U j ) = τ j Σ. II) For each point x ∈ N there is an open neighbourhood U x =: U of x, a number τ x =: τ > 0 and a bi-Lipschitz mapping φ x =: φ from an open neighbourhood of U into R n , such that φ(x) = 0 ∈ R n , φ(U ) = τ K, φ(U ∩ Ω) = τ K and φ(∂Ω ∩ U ) = τ Σ. a) If x ∈ N , then U does not intersect D, i.e., φ(D ∩ U ) = ∅. b) If x ∈ N ∩ D, then φ(D ∩ U ) = τ Σ 0 . III) Each of the occurring mappings φ is, in addition, volume-preserving.
a Figure 2. The -topologically regularized -double beam is the the prototype of a domain which is Lipschitzian, but not strongly Lipschitzian. Moreover, a boundary chart around the point a may be constructed also as a volume-preserving one, cf. [33,Ch. 7].

Remark 2.5. i) Primarily, Assumption 2.4 gives a typology of boundary points
of Ω in the following sense: I) sets the conditions for points from the relative interior of the Dirichlet part, while IIa) is a condition for the Neumann boundary points and IIb) gives a condition for points from the border between Dirichlet and Neumann boundary part. In fact, this latter condition goes back to the paper of Gröger [28]. A simplifying topological characterization of Gröger's condition in case of space dimensions n = 2 and n = 3 is given in [32,Ch. 5]. ii) Note that Assumption 2.4 I) in particular demands that every connected component V j of U ∩ Ω satisfies the assumptions for the Dirichlet boundary part of a Lipschitz domain on its own. Setting V := U ∩ Ω in II), we find ∂V ∩ U = ∂(Ω ∩ U ) ∩ U = ∂Ω ∩ U , which is the analogue to ∂V j ∩ U j in I) and shows compatibility of the conditions on the mappings on τ Σ in I) and II). iii) The inclusions ∂V j ⊂ V j ⊂ U j imply the disjoint union ∂V j = (∂V j ∩ U j ) ∪ (∂V j ∩ ∂U j ). Thus, ∂V j ∩ U j is the relevant part of ∂V j . Moreover, it is indeed not really necessary to demand the properties φ(U ∩ ∂Ω) = τ Σ and φ j (∂V j ∩ U j ) = τ j Σ -they follow from the other ones by purely topological reasons. We have added them only to be at this point more suggestive, see also the previous item of this remark. iv) In particular, all domains with Lipschitz boundary (strong Lipschitz domains) admit bi-Lipschitzian boundary charts which are volume preserving: if, after a shift and an orthogonal transformation, the domain lies locally beyond a graph of a Lipschitz function ψ, one defines φ(x 1 , . . . , x n ) = (x 1 − ψ(x 2 , . . . , x n ), x 2 , . . . , x n ).
This way, the mapping φ obviously is bi-Lipschitz and the determinant of its Jacobian is identically 1. v) Note that the additional property volume-preserving also has been required in several similar contexts (see [25] and [29]). It turns out that the property bi-Lipschitz together with volume-preserving is not a too restrictive condition. In particular, there are bi-Lipschitzian, volume-preserving mappings -although not easy to construct -which map the ball onto the cylinder, the ball onto the cube and the ball onto the half ball, see [24], see also [19]. The general message is that this class of transformations has enough flexibility to map "non-smooth" objects onto smooth ones.
In the following, all considered space are real ones.
Definition 2.6. Let Λ be a bounded open set, and let F be a closed part of ∂Λ. For 1 ≤ q < ∞ we define W 1,q F (Λ) as the closure of in the real Sobolev space W 1,q (Λ). If F = ∂Λ, then we write W 1,q F (Λ) = W 1,q ∂Λ (Λ) = W 1,q 0 (Λ). If Λ is a Lipschitz domain and F = ∅, then W 1,q F (Λ) equals the usual Sobolev space W 1,q (Λ). The latter follows from the fact that, for Lipschitz domains, the set C ∞ ∅ (Λ) is dense in W 1,q (Λ), cf. We call | · | κ,O the Hölder seminorm. x−y κ for one ε > 0 under control in order to show Hölder continuity. Namely, one has for x, y ∈ O with x−y ≥ ε the trivial estimate |ψ(x)−ψ(y)| ≤ 2 ε κ sup |ψ| x−y κ . iv) The reader should carefully notice that in [40,Ch. I.1] there are two notions of Hölder continuity in use, one coinciding with ours.
Furthermore, for the sake of clarity, we will write ·, · X for the dual pairing of elements of X and its dual X . For a (vector-valued) function u, defined on J, we denote by u its derivative in the sense of vector valued distributions, cf. [4,Ch. III.1] and define W 1,s (J; X) := {v : v, v ∈ L s (J; X)}.
The symbol ∇ always stands for the spatial gradient -even if the corresponding function depends on space and time.
Definition 2.9. Let Λ be a bounded domain, and let F ⊆ ∂Λ be closed. Let ρ : Λ → M n be a bounded Lebesgue-measurable function. Then we define −∇ · ρ∇ + 1 : 3) We maintain the notation of the operator when the range space is restricted to W −1,q F (Λ) for q > 2. By Hölder's inquality, the domain of this restricted operator always contains the space (Ω)), with the analogous restriction conventions for the spatial operator as for the timeindependent case. Proposition 2.10. Suppose that V → H → V is a Gelfand triplet of real Hilbert spaces with dense embeddings. Let {a t } t∈J be a family of bilinear forms on V the norms of which are uniformly bounded and such that each a t is coercive with a coercivity constant κ, also uniformly in t ∈ J. Suppose that the mapping J t → a t (ψ, ϕ) is measurable for all ψ, ϕ ∈ V . Then, for any f ∈ L 2 (J; V ), there is a unique u = u f ∈ L 2 (J; V ) ∩ W 1,2 (J; V ) such that u(T 0 ) = 0 and holds true for almost all t ∈ J. Moreover, u admits the following estimates: Thus, the mapping which assigns to the right hand side f ∈ L 2 (J; V ) the solution u of (2.4) with initial value u(T 0 ) = 0 is well-defined and continuous from L 2 (J; V ) into L 2 (J; V ) ∩ C(J; H), and its norm is not larger than 1 κ + 1 κ .
Remark 2.11. Defining, for t ∈ J, the operator A(t) : V → V by for almost almost all t ∈ J.
In the following considerations using Proposition 2.10, the spaces W 1,2 F (Λ) always play the role of V , and the form a t will be of type for some bounded coefficient function σ : J × Λ → M n . Clearly, the resulting operator A(t) is then the corresponding divergence operator −∇ · σ(t, ·)∇ + 1 on W 1,2 F (Λ). Note that, vice-versa, −∇ · σ(t, ·)∇ + 1 also induces a family of forms a t on W 1,2 F (Λ) × W 1,2 F (Λ). Remark 2.12. Let us point out that the following considerations may also be carried out for the operators −∇·σ(t, ·)∇ alone, if H n−1 (F ) = 0. The corresponding form (as in (2.8)) is then, via the Poincaré-inequality, still coercive on W 1,2 F (Λ) (see [6,Rem. 3.4] for the Poincaré-inequality, see also Theorem 4.3), while the rest of the considerations remains untouched in its essence.
The subsequent theorem contains the main result of this paper.
Theorem 2.13. Assume that Ω and D are given and fulfil Assumption 2.4, and let µ ∈ M n (κ 0 , κ 1 ) for some κ 0 , κ 1 > 0. Let q > n and s > 2(1 − n q ) −1 be fixed and f ∈ L s (J; W −1,q D (Ω)). Then the solution u = u f of the equation in the sense of Proposition 2.10/Remark 2.11 exists and is unique. Moreover, let B denote the unit ball in L s (J; W −1,q D (Ω)). Then the following holds true: i) The supremum sup f ∈B u f L ∞ (J×Ω) is finite and depends only on κ 0 , κ 1 , q and s. ii) There is an α > 0, such that even sup f ∈B u f C α (J×Ω) is finite and depends exclusively on κ 0 , κ 1 , q and s. In other words: Let (∂ t + A(µ)) −1 denote the linear operator which assigns to the right-hand side of the parabolic equation in (2.9) the solution u = u f with initial value u 0 = 0. Then the mapping is well-defined and continuous for some α. For fixed κ 0 , κ 1 , the mappings (2.10) are equicontinuous for all coefficient functions µ ∈ M n (κ 0 , κ 1 ).
Remark 2.14. It is straight-forward to check that for q, s ≥ 2, L s (J; In this sense, right-hand sides f from L s (J; W −1,q D (Ω)) are implicitly always to be understood as right-hand sides from L 2 (J; W −1,2 D (Ω)) without further comment in the sequel.
3. The proof. Let us give the proof of Theorem 2.13. We first collect some classical results of Ladyzhenskaya et al. [40] adopted for our cause. The basis of our considerations will be Corollaries 3.5 and 3.7 which are based on space-time local estimates for so-called generalized solutions of corresponding equations in [40,Ch. III]. However, in order to use those, we invest quite some work and introduce a non-trivial localization-procedure for (2.9) which allows to transform the localized equation onto a very regular object, namely the lower half-cubes τ K and (via reflection) the full cubes τK in such a way that the resulting equation still provides a generalized equation in the sense of Ladyzhenskaya.
3.1. Classical results. We begin by introducing the notion of a generalized equation. The crucial link to the concept of Lions is the space V 1,0 2 (J × Ξ) introduced in the next definition, which corresponds to the spaces L 2 (J; V ) ∩ C(J; H) in Proposition 2.10.

Remark 3.2.
Integrating the term T T0 Ξ u ∂ϑ ∂t dxdt formally by parts with respect to time, the term Ξ u(T 0 , x)ϑ(T 0 , x) dx appears, which is not compensated by other terms in (3.2). Thus, if test functions ϑ are admitted which are nonzero on {T 0 }×Ξ, such as those from W 1,2 J×∂Ξ (J × Ξ), this enforces u(T 0 , ·) to be the zero functionon a formal level.
The next results are in their essence space-time local estimates for generalized solutions if the right-hand side in (3.1) is regular enough. However, for initial value 0 we may re-obtain the estimates for the whole time interval J, see Corollaries 3.5 and 3.7. Proposition 3.3. [40, Ch. III, Thm. 8.1] Let Ξ ⊂ R n be a bounded Lipschitz domain, and let µ be from M n (κ 0 , κ 1 ). Fix q > n and s > 2(1 − n q ) −1 . Let the set F be given such that for some C ≥ 0. Moreover, assume that for every f ∈ F a generalized solution u = u f of (3.1) exists and {u f : f ∈ F} is contained in a ball around 0 in V 1,0 2 (J ×Ξ) with radius r V . i) Let Ξ 0 ⊂ Ξ be a subdomain which has a positive distance d < T 1 − T 0 to ∂Ξ.
ii) Let F be a closed part of ∂Ξ and let all u f belong to the space L 2 (J; W 1,2 F (Ξ)). If a subdomain Ξ 0 of Ξ has a positive distance d < T 1 − T 0 to ∂Ξ \ F , then also sup f∈F u f L ∞ (]T0+d,T1[×Ξ0) is finite, and depends only on n, κ 0 , κ 1 , r V , d, q, s and C.
Proof. One associates to the problem (3.1) another one on the interval J 0 := ]T 0 − d − 1, T 1 [ in the following manner: one defines a coefficient functionμ on J 0 × Ξ by else.
Moreover, one defines a new right-hand sidef as 0 on J 0 \ J and as f on J and finds the solutionǔ on J 0 ×Ξ with u(T 0 −d−1) = 0. This solutionǔ is zero on (J 0 \J)×Ξ and coincides with u on J × Ξ. Applying Proposition 3.3 i) to the functionǔ one gets i). Point ii) is deduced analogously from ii) of the foregoing Proposition. , and suppose µ ∈ M n (κ 0 , κ 1 ). Fix q > n and s > 2(1 − n q ) −1 . Assume that F is again a subset of the set in (3.3), such that for every f ∈ F a generalized solution u = u f of (3.1) exists and that this set of generalized solutions is contained in a ball around 0 in L ∞ (J × Ξ) with radius r ∞ . Then there is an α > 0 such that the following is true: is finite and depends only on n, κ 0 , κ 1 , r ∞ , d, q, s and C, cf. (2.1) and (3.3). ii) Let F be a closed part of ∂Ξ and suppose that all u f belong to the space L 2 (J; W 1,2 F (Ξ)). If a subdomain Ξ 0 of Ξ has a positive distance d ∈ ]0, T 1 − T 0 [ to ∂Ξ \ F , then the supremum sup f∈F u f C α (]T0+d,T1[×Ξ0) is finite and depends only on n, κ 0 , κ 1 , r ∞ , d, q, s and C.

HANNES MEINLSCHMIDT AND JOACHIM REHBERG
Corollary 3.7. Suppose the assumptions of Proposition 3.6 to hold and assume, additionally, that the initial value u 0 of the solution is zero. Then there is an α > 0 such that the following is true: i) For every subdomain Ξ 0 ⊂ Ξ having a positive distance d to the boundary ∂Ξ, is finite and depends only on n, κ 0 , κ 1 , r ∞ , d, q, s and C, cf. (2.1) and (3.3). ii) Let F be a closed part of ∂Ξ and suppose that each u f belongs to the space L 2 (J; W 1,2 F (Ξ)). Then, for any subdomain Ξ 0 with a positive distance d to ∂Ξ \ F , sup f∈F u f C α (]T0,T1[×Ξ0) is finite and depends only on n, κ 0 , κ 1 , r ∞ , d, q, s and C.
The proof works analogously to the one of Corollary 3.5.
Remark 3.8. In fact, the quoted result holds for much more general domains as convex ones. However, we have good reasons to restrict ourselves to this case: • If Ξ is convex and B ⊂ R n is a ball, then Ξ ∩ B is still convex and therefore always consists of only one component. Thus, one may deal with the classical notion of Hölder continuity -and not of the much more sophisticated one in [40,Ch. I] • Secondly, if Ξ is convex, then every point x ∈ ∂Ξ admits a supporting hyperplane such that Ξ lies on one side of this hyperplane. Thus, for any ball B ⊂ R n with center x, the intersection Ξ ∩ B has at most half the measure of B, what makes the crucial "Condition A" ([40, Ch. 1, p.9]) obviously fulfilled in our context, with the constant θ 0 = 1 2 -universal for all convex domains and all balls.
• We will need the result only in case of balls, cubes and half cubes, serving as our local model sets.
The next proposition establishes the link between generalized solutions and solutions in the sense of Proposition 2.10. For doing so, we restrict ourselves to the case of right hand sides which are step functions in time only (these being dense in the whole space under consideration). The reason is as follows: By a classical theorem, the elements f from W −1,q D (Ω) may be represented as the sum of the divergence of a R n -valued function f ∈ L q and f itself. The problem is that this representation is highly non-unique and, the worse, not obviously linear. So we preferred to restrict ourselves to step functions and to use the corresponding representation theorem separately on any of the constancy intervals only. Proposition 3.9. Let Ξ ⊂ R n be a bounded Lipschitz domain, and F be a closed portion of the boundary ∂Ξ.
A proof of this is given in the Appendix.
3.2. Preliminaries. One of the main technical ingredients of our proof is a certain localization procedure of the equation (2.9). In contrast to [28] and many following papers it is not carried out by multiplying the solution with suitable cut-off functions and afterwards deriving a corresponding equation for the product. We only restrict the function to open subsets of the domain and deduce a corresponding equation for this restriction -in an adequate weak formulation. In fact, this idea was developed in [17] for elliptic problems.
The following lemmata allow us in the sequel to perform this in an appropriate manner. The first lemma covers the cases of neighbourhoods of interior points of Ω and from the Neumann boundary (i.e., satisfying case II) of Assumption 2.4).
. Thus, the restriction operator from W 1,p D (Ω) is a continuous one into W 1,p R (Λ) with norm not larger than 1. Proof. For i) and ii), see [17,Lem. 6.13]. iii) Observe that D ∩ U ⊆ ∂Ω ∩ U ⊆ ∂Λ. Since ∂Λ is closed, this gives R ⊆ ∂Λ. On the other hand, . In case I) in Assumption 2.4 the local model set is allowed to be disconnected. Nevertheless, one can also in this case find an adequate localization procedure. In the spirit of Remark 2.5, this relies on the localization procedure for each of the connected components. i) There is an isometric operator E j which extends any function from W 1,p 0 (V j ) by 0 to a function from Proof. i) The support of every function from C ∞ 0 (V j ) has a positive distance to ∂Ω; thus the extension by zero leads to a function from C ∞ 0 (Ω) in this case. The general claim follows by density. ii) By the definition of V j it is clear that ∂V j is contained in U j ∩ Ω. Now suppose that a point y ∈ ∂V j lies in U j ∩ Ω (i.e., not on ∂(U j ∩ Ω)). Since U j ∩ Ω is open, we find an open ball B containing y which is still a subset of U j ∩ Ω. By supposition, y is a boundary point of V j , hence V j ∩ B = ∅. Thus, the connectedness of both V j and B implies that V j ∪ B ⊃ V j is also open and connected -and, hence, identical with V j . But then B ⊂ V j which is a contradiction to y being a boundary point of We aim lastly at equations on τ K and τ K for localized equations in neighbourhoods of boundary points of Ω, to be achieved via the bi-Lipschitzian transformations occurring in Assumption 2.4. Hence it is, of course, of interest onto which sets the different boundary parts are mapped by these transformations: • If x satisfies Assumption 2.4 I), then for each j ∈ {1, . . . , k} one has φ j (∂V j ) = ∂(τ j K) and, in the terminology of Lemma 3.11, • If x satisfies Assumption 2.4 II), one has in the terminology of Lemma 3.10 (putting U := U x and φ : Proof. This is straight-forward from the mapping properties of the transformations φ x and φ j . It turns out that the model constellation in Assumption 2.4 IIb) is indeed suggestive, but not optimal for further analytical purpose. We show in the next lemma that it can be replaced by another one which is much more controllable later, cf. [32,Sect. 4.2].
Lemma 3.13. For every τ > 0, there exists a volume-preserving, bi-Lipschitzian mapping ς n : Let us start with the case n = 2, thereby focussing first on the case τ = 1. We define on the lower halfspace {(x, y) ∈ R 2 : y ≤ 0} Observing that ξ 1 acts as the identity on the x-axis, we may define ξ 1 on the upper half space {(x, y) ∈ R 2 : y > 0} by ξ 1 (x, y) = (x, y/2). In this way we obtain a globally bi-Lipschitz transformation ξ 1 from R 2 onto itself that transforms K ∪ Σ 0 onto the triangle shown in Figure 3. Next we define the bi-Lipschitz mapping in order to get the geometric constellation in Figure 4. If ξ 3 is the (counter- clockwise) rotation of π/4 around 0 ∈ R 2 , we thus have achieved that ξ := ξ 3 ξ 2 ξ 1 : R 2 → R 2 is bi-Lipschitzian and satisfies The assertion for K is verified by a straight forward calculation. As is easy to check, the determinant of the Jacobian is identically one almost everywhere. Hence, ς 2 is volume-preserving.
If τ = 1, then one first applies the homothety y → 1 τ y, then the mapping ς 2 just constructed for the case τ = 1 and afterwards the inverse homothety y → τ y.
Corollary 3.14. Suppose that Assumption 2.4 IIb) holds true. Then for every point x from ∂D (within ∂Ω) there is a an open neighbourhood U x , a positive number τ = τ x and a bi-Lipschitzian, volume-preserving mapping from a neighbourhood of Proof. If one defines the asserted mapping as the composition ς n • φ x , then the application of Lemma 3.12 and Lemma 3.13 gives the assertion.
Having the bi-Lipschitz mappings φ and ς defined above at hand, we collect properties of bi-Lipschitzian transformations if applied to the typical data of parabolic equations as (2.9). It turns out that (volume-preserving) bi-Lipschitz mappings essentially preserve the structure of the underlying problem.
Proposition 3.15. Let Λ be a bounded Lipschitz domain, and let F be a closed portion of its boundary. Assume that ζ is a bi-Lipschitzian mapping from a neighbourhood of Λ into R n . Define for any function ϕ : i) For every ϕ ∈ W 1,1 (ζ(Λ)), the (generalized) gradient of the function ϕ • ζ is calculated for almost all x ∈ Λ as follows: ii) For every p ∈ ]1, ∞[, the mapping Φ induces linear, topological isomorphisms and C α (Λ). The norms of Φ 0,α and Φ −1 0,α only depend on the Lipschitz constants of ζ and ζ −1 . iv) Let ρ : Λ → M n be bounded and measurable. Then one has for every p ∈ ]1, ∞[ and every pair (ψ, for almost all y ∈ ζ(Λ). Here, Dζ denotes the Jacobian of ζ and det(Dζ) the corresponding determinant. v) Let l ζ , l ζ −1 denote the Lipschitz constants of ζ and ζ −1 , respectively, and assume that ζ is volume preserving. If ρ takes its values in M n (κ 0 , κ 1 ), then ρ ζ takes its values in M n ( κ 0 , κ 1 ) where κ 0 := κ0 iii) is obvious. Assertion iv) can be deduced from i), for a complete proof see [31,Prop. 16]. v) First one observes that for a volume-preserving mapping ζ the function | det(Dζ)(·)| is identically 1, [18,Ch. 3]. Secondly, Rademacher's theorem shows that With all this in mind, one easily calculates for almost all y ∈ ζ(Λ) and all z ∈ R n as follows: In order to deduce the lower bound, one first recalls the equality 3.3. Localization, transformation, reflection. Now we have the principle ideas at hand and will first localize the parabolic equation suitably in order to consider it on smaller sets. The resulting equations are then transformed by bi-Lipschitzian mappings, corresponding of course to Assumption 2.4, to equations on the half cube K. In the case of points from the Neumann boundary part, one finally needs a reflection argument, which will be established in the last part of this subsection. Having this in mind, let us now localize the equation (Ω)). Note that a solution u to this equation belongs to the space (Ω)) → C(J; L 2 (Ω)), cf. Proposition 2.10. Let us fix an arbitrary point x ∈ Ω and consider an open neighbourhood U of x. If x ∈ Ω, we assume U ⊂ Ω. We will now localize the equation around x according to the constructions from Lemmata 3.10 (for the first two cases) and 3.11 (the last case), respectively: . . , k}. The following localization procedure then has to be done for every j ∈ {1, . . . , k}. We will, however, omit the index j to simplify the notation.
In this terminology, one calculates for w ∈ W 1,2 D (Ω) and every ϕ ∈ W 1,2 Remark 3.17. The first term in (3.9) does not contain abuse of the above introduced notation in the following sense: for w ∈ W 1,2 D (Ω) the restriction w| Λ belongs to the space W 1,2 R (Λ), cf. Lemma 3.10 iii) and 3.11 iii). The operator Assume now that a given function u ∈ L 2 (J; (Ω)-valued distributions on the righthand side. Note carefully that everything is indeed in order since E U : is well-defined and continuous, thanks to Lemma 3.10 ii) and Lemma 3.11 i). One step further, using (3.10) and (3.9) in case of w = u(t) and ρ = µ(t, ·), one obtains for every ϕ ∈ W 1,2 E (Λ) and almost everywhere on J since E U is an isometry. This shows the following: the function J t → f U (t), defining the right-hand side in (3.11), belongs to L 2 (J; W −1,2 E (Λ)), and its norm does ) with a similar estimate. In this spirit, let us write (3.11) in the form Remark 3.18. In any case, the property u ∈ L 2 (J; W 1,2 D (Ω)) ∩ C(J; L 2 (Ω)) implies that we have u| Λ ∈ L 2 (J; W 1,2 R (Λ))∩C(J; L 2 (Λ)), and the corresponding V 1,0 2 -norm of u| Λ is not larger as the V 1,0 2 -norm of u, cf. Lemmata 3.10 and 3.11. This completes the localization procedure so far: For every possible constellation in and around a point x ∈ Ω, we have constructed a suitable local equation in W −1,2 E (Λ) which is satisfied by the global solution u. Next, we transform these local equations according to Assumption 2.4 using the properties of the transformations established in Proposition 3.15. Suppose from now on that for every point x ∈ ∂Ω, a neighbourhood U of x is given as declared in the fitting case in Assumption 2.4 and that Λ, E and R are chosen accordingly as in the localization procedure above (with the obvious adjustments).
We now exploit III) of Assumption 2.4, that is, for each case of boundary points x, there is a volume-preserving, bi-Lipschitzian mapping ζ from a neighbourhood of Λ onto a neighbourhood of the cube τ K. Let us assume that E is mapped onto E • ⊂ ∂(τ K), and that R is mapped onto R • ⊂ ∂(τ K) -where ζ and E • , R • will be specified later and, of course, in correspondence with Assumption 2.4, Lemma 3.12 and Corollary 3.14.
On the other hand, one gets for every ϕ ∈ W 1,2 (3.13) leads to the following equation for the transformed function v: the constant c only depending on ζ, see Proposition 3.15i). Thus, for almost every t ∈ J, the linear form . If one denotes this linear form by g(t), then (3.16) shows the following: if f in (2.7), cf. also (2.9), even belongs to L s (J; W −1,q D (Ω)), then g is from L s (J; W −1,q E• (τ K)) and, additionally, fulfils the estimate the constant c only depending on the mapping ζ. Expressing the right-hand side of (3.15) in this manner, we get the final equation for v on τ K, namely Remark 3.19. Again, the property u| Λ ∈ L 2 (J; W 1,2 R (Λ) ∩ C(J; L 2 (Λ)) leads to v being from L 2 (J; W 1,2 R• (τ K)) ∩ C(J; L 2 (τ K)) inclusively a corresponding estimate -where the norm depends only on the bi-Lipschitz mapping ζ, cf. Lemma 3.15 ii). Moreover, (3.14) gives the inclusion v ∈ W 1,2 (J; W −1,2 E• (τ K)) together with estimates for the corresponding norms.
Let us now specify the mapping ζ in dependence of the different cases in Assumption 2.4 and the conventions from the beginning of the localization procedure, defining the sets E • = ζ(E) and R • = ζ(R) correspondingly: • In case I) one puts ζ j := φ j , thus obtaining for each j ∈ {1, . . . , k}, see Lemma 3.12. • In case IIa), we set ζ = φ x , such that cf. Lemma 3.12. • In case IIb) we choose ζ := ς n • φ x and obtain, in view of Corollary 3.14, Observe that in this last case ζ(x) = (0, . . . , 0, −τ, 0). Having the transformed equations on the half cubes with transformed boundary conditions at hand, we lastly introduce reflection for case II) from Assumption 2.4. Inspection of Corollaries 3.5 and 3.7 reveals why this is necessary: Both corollaries require a subdomain Ξ 0 which has a positive distance to the whole boundary ∂Ξ or to the complement of the Dirichlet boundary part F . But in case II) of Assumption 2.4, after the localization and transformation procedure we end up with ζ(x) being a boundary point on the half square without prescribed Dirichlet boundary part (remember ζ(R) = ∅ in case a)) and ζ(x) being at the boundary of the Dirichlet boundary part itself, respectively. Both cases do not admit a suitable neighbourhood of ζ(x) which would satisfy the assumptions of Corollaries 3.5 and 3.7. By reflecting the equation across the "upper" plate of the half cubes, we obtain ζ(x) being inner points of the whole cube and the (combined) Dirchlet boundary part, respectively, allowing to use the aforementioned corollaries.
Let us first define for x = (x 1 , . . . , x n ) ∈ R n the symbol x − := (x 1 , . . . , x n−1 , −x n ), and for a n × n matrix , the matrix − by − i,j := if i, j < n, − i,j , if i = n and j = n or j = n and i = n, Corresponding to a coefficient function ρ on τ K, we then define the coefficient functionρ on τ K byρ Finally, we define for w ∈ L 1 (τ K) the function w − by w − (x) = w(x − ), and for w ∈ L 1 (τ K) the (symmetrically) reflected function by Lemma 3.4]. Lastly, standard arguments show that Ew may be approximated in the W 1,p -norm by restrictions of C ∞ 0 (R n )-functions the support of which avoids F . Let us next introduce an extension operator for distribution-type objects: For p ∈ ]1, ∞[, define the extension operator S : . We immediately obtain the following properties: Proof. One has for all ϕ ∈ W 1,p F (τ K) the identity which proves the first point. Moreover, the operator under consideration is the adjoint of the continuous operator W 1,p , which implies both assertions from the second point.
) and still has initial value 0. Finally, the function Ev satisfies , and the norm of Sg is not larger than two times the norm of g.
Proof. i) The assertion is obtained by the definitions of Ew, Sh, ∇ · ρ∇, ∇ ·ρ∇ and straightforward calculations, based on Proposition 3.15, when applied to the transformation x → x − . ii) The first two assertions follow from Lemmata 3.20 and 3.21; let us show that Ev indeed satisfies the correct equation: coming from (3.24), for ψ ∈ W 1,2 E• (K) is for almost all t ∈ J an equation of type (3.22). According to i), this leads to an equation what gives the last assertion. iii) The assertion follows immediately from Lemma 3.21 ii).

3.4.
The core of the proof. Now we have all preparations at hand and will prove our main result, Theorem 2.13. The following lemma is the starting point for the usage of the foregoing results. (Ω)). Then, for every f ∈ B, the solution u = u f of (2.4)/ (2.7) is contained in a ball B around 0 in V 1,0 2 (Ω) with radius here κ = min(κ 0 , 1) being the (uniform) coercivity constant of the forms Hence, for all coefficient functions µ admitting the same ellipticity constant κ 0 , in particular all those from M(κ 0 , κ 1 ), the radii r V may be taken uniformly.
Proof. The unit ball B is contained in the corresponding ball in L 2 (J; W −1,2

D
(Ω)) with radius |Ω| To this end, we localize the parabolic equation (2.9) with respect to a suitable neighbourhood of each point, transform the localized equations to such on the half cubes and reflect the problem to the whole cube, if necessary. This allows to use Corollaries 3.5 and 3.7, respectively, to deduce the wished-for estimates.
Choose for any point x ∈ Ω a ball B • x around x which satisfies B • x ⊂ Ω and has a positive distance to ∂Ω. Define B x as the ball with half the radius of B • x . Further, for every y ∈ ∂Ω, let U y be an open neighbourhood of y which satisfies the conditions in Assumption 2.4. In case I of that assumption, we put W y = ∩ j φ −1 j ( τj 2 K). If y fulfils case II of that assumption, then we put W y = φ −1 y τy 2 K , which implies W y ∩ Ω = φ −1 y τy 2 K . Obviously, the collection of the sets {B x } x∈Ω and {W y } y∈∂Ω forms an open covering of Ω. Let B x1 , . . . , B xm 0 , W y1 , . . . , W ym 1 be a finite subcovering.
Before we continue, we need the following property of the sets W y in case of Assumption 2.4 I):

28)
the right hand side being a disjoint union.
Let B be again the unit ball in L s (J; W −1,q D (Ω)), and let B step denote the set of step functions in B.
Step 2. We consider the restricted problem on each of the balls B • x l according to Ch. 3.3 (there setting U = B • x l ), cf. (3.13), where the right-hand side f U in the restricted problem is still bounded by 1 for f ∈ B. For f ∈ B step , however, the solution u f is a generalized solution of a corresponding generalized problem on B • x l with right-hand side f U , cf. Proposition 3.9, f U still being a step function in time and contained in the ball with radius 2 in L s (J; L q (B x l ; R n+1 )). Thanks to Corollary 3.5, the functions u f | J×Bx l are essentially bounded, and the norms u| J×Bx l L ∞ (J×Bx l ) are bounded uniformly in f ∈ B step and in µ ∈ M n (κ 0 , κ 1 ). This of course implies uniform boundedness for all l ∈ {1, . . . , m 0 }.
Step 3. Let us now consider the boundary points, thereby temporarily fixing y = y l ∈ ∂Ω.
We start with case I) of Assumption 2.4): Intersecting Ω with U y , the restriction of the function u = u f to each of the connected components V j belongs to W 1,2 Rj (V j ) when taking R j as ∂V j ∩ U j , cf. Lemma 3.11. One obtains a restricted problem on V j which is of the same quality as (2.9), cf. (3.13) with Λ = V j and E = ∂V j . Further, we transform this resulting problem to a problem for the function v j := u| Vj •φ −1 j on τ j K. According to (3.18)/(3.19), one ends up with an equation for the transformed function v j on τ j K with new right-hand side g j ∈ L s (J; W −1,q 0 (τ j K)), which is still a step function in time. By Proposition 3.9, v j is then a generalized solution of the transformed equation (3.18) on τ j K with right-hand side g j ∈ L s (J; L q (τ j K; R n+1 )) and coefficient function µ φj . This is the setting for all j ∈ {1, . . . , k}. Let us show that we are in the situation to use Corollary 3.5 for each problem on V j .
• The new right-hand sides g j may be estimated suitably with respect to the original ones, cf. (3.17) and Proposition 3.9, giving g j L s (J;L q (τj K;R n+1 )) ≤ 2 g j L s (J;W −1,q 0 (τj K)) ≤ 2c j . • The resulting transformed coefficient functions µ φj on J × τ j K still admit uniform upper boundsκ 1,j , and uniform ellipticity constantsκ 0,j , cf. Proposition 3.15 v). • Moreover, it is clear that u| J×Vj V 1,0 2 (J×Vj ) is not larger than u V 1,0 2 (J×Ω) , which was uniformly bounded over M n (κ 0 , κ 1 ) and with respect to f ∈ B by the constant r V thanks to Lemma 3.23. Proposition 3.15 ii) shows that v j V 1,0 2 (J×τj K) may be estimated byc j r V for some constantc j depending on j via φ j . • By Remarks 3.18 and 3.19, we have v j ∈ L 2 (J; W 1,2 τj Σ (τ j K)). Summing up, we have, for each j, coefficient functions from M n (κ 0,j ,κ 1,j ) and right-hand sides g j contained in the 2c j -ball around 0 in L s (J; L q (τ j K; R n+1 )) such that the generalized solutions v j to all those right-hand sides are in turn contained in a ball with radiusc j r V in V 1,0 2 (J × τ j K) and even belong to L 2 (J; W 1,2 τj Σ (τ j K)). Applying Corollary 3.5 ii) with the subdomain τj 2 K, we get L ∞ -bounds on J × τj 2 K for every v j which are uniform in f ∈ B step and µ ∈ M n (κ 0 , κ 1 ). Thanks to (3.28), this gives L ∞ -bounds for u on J × (W y ∩ Ω), uniformly for f ∈ B step and µ ∈ M n (κ 0 , κ 1 ).
Next we will consider the case II) in Assumption 2.4. We abbreviate τ y =: τ . Localizing around y with respect to U y according to Ch. 3.3 results in a problem for u| Λ in the form (3.13) with Λ = U y ∩ Ω and E = ∂Λ \ (N ∩ U y ). By afterwards transforming the resulting problem via ζ = φ y (case IIa)) and ζ = ς n •φ y (case IIb)), one again ends up with a problem on τ K as in (3.18), which we interpret as a generalized problem solved by the function v = u| Λ • ζ. We obtain analogous estimates and bounds, especially uniformly in µ ∈ M n (κ 0 , κ 1 ) and f ∈ B step , for the coefficient function µ φy , right-hand side g and solution v ∈ V 1,0 2 (J × τ K) as we did for each j in the previously handled case I). The following considerations require further distinguishing the assumptions.
In case IIa) of Assumption 2.4, we get v ∈ L 2 (J; W 1,2 (τ K)) according to Remark 3.19 and (3.20). Here, the upper plate τ Σ is disjoint to the (transformed) Dirichlet boundary part (which in fact is even empty here, cf. (3.20)), permitting the direct application of Corollary 3.5 for a neighbourhood of φ y (y) = 0, since the latter is obviously also a boundary point of τ K. However, we may reflect the problem across τ Σ according to Lemma 3.22, thus obtaining the corresponding equation (3.25) on τ K for the symmetrically reflected function Ev. It is clear that the bounds for the data and the V 1,0 2 -estimate for v carry over to τK in a straight forward manner, cf. Lemma 3.21 and the definition of the reflection operator E, and that φ y (y) = 0 ∈ τ K is an interior point in τ K. Hence, we may apply Corollary 3.5 i) for the subdomain τ 2 K and obtain an L ∞ -bound for Ev on J × τ 2 K, again uniformly in µ ∈ M n (κ 0 , κ 1 ) and f ∈ B step . Obviously, this implies an L ∞ -bound with the same property for u on J × (W y ∩ Ω) = J × φ −1 ( τ 2 K).
and one observes that τ K has the distance τ 2 to the set Another application of Corollary 3.5 ii), this time for the subdomain τ K, gives an L ∞ -bound for v on J ×τ K, and, correspondingly, on J ×ζ −1 (τ K) = J ×φ −1 y ( τ 2 K) = J × (W y ∩ Ω), again uniformly for f ∈ B step with respect to κ 0 , κ 1 .
Hence we have L ∞ -bounds on J × W y l ∩ Ω for each l ∈ {1, . . . , m 1 } which then clearly implies L ∞ -bounds uniform in l. Since the finite system B x1 , . . . , B xm 0 , W y1 ∩ Ω, . . . , W ym 1 ∩ Ω is an open covering of Ω, this altogether gives L ∞ -bounds on the whole set J × Ω, which are uniform for all f ∈ B step for the corresponding functions u f and which do only depend on the constants κ 0 , κ 1 . This was the first point of Theorem 2.13.
Step 4. Having the essential boundedness at hand, we will now establish the Hölder estimates by essentially re-iterating the considerations in the foregoing steps, this time investing the obtained uniform global L ∞ -bounds instead of the V 1,0 2 -estimates and then applying Corollary 3.7 instead of Corollary 3.5.
In detail: Both Step 2, which was the case of the balls B x l , and the considerations in case II) of Assumption 2.4 in Step 3 work exactly as above, using Corollary 3.7 this time. In case I) of Step 3, the situation is a bit more complicated and needs more care: Repeating the procedure outlined above to the point where Lemma 3.24 and (3.28) are used, one obtains the Hölder property for every transformed local solution v j (including estimates uniform in f ∈ B step , depending only on κ 0 , κ 1 ) on the set J × τj 2 K for each j ∈ {1, . . . , k}. Due to the disjoint union in (3.28), u can be represented as u = k j=1 v j • φ j on J × (Ω ∩ W y ). It is essential to observe, however, that this implies only Hölder continuity for u on each of the disjoint sets J × φ −1 j ( τj 2 K) ⊂ J × V j on its own -it is not (yet) clear why the Hölder property should hold "across" different connected components. Let us note that this is exactly the result of Ladyshenskaya in [40]. In the sequel we will show that our setting allows to derive from this the required global Hölder estimates on the sets J × (Ω ∩ W y ).
Let us in the following identify the Hölderian function v j , defined on J × τj 2 K, with its unique Hölderian extension on J × τj 2 K, cf. Remark 2.8. The crucial point is here that we imposed in our general ansatz a very special boundary value on the whole Dirichlet part D of the boundary -namely, 0. Indeed, the property v j ∈ L 2 (J; W 1,2 τj Σ (τ j K)) implies that v j (t, ·) has trace 0 on τj 2 Σ, i.e., vanishes there almost everywhere with respect to the boundary measure H n−1 for almost all t ∈ J, see Remark 3.4. However, v j (t, ·) is also a continuous function on τj 2 K, and τj 2 K has a Lipschitz-boundary around 0, hence in fact v j (t, ·) ≡ 0 on τj 2 Σ for almost all t ∈ J. But then, this time due to continuity in time, v j must be identically 0 on the whole J × τj 2 Σ. It is straight forward to verify that the continuationv j of v j to J × τj 2 K by zero is also Hölder continuous -with the same Hölder-norm as v j on J × τj 2 K. This means we may extend u viaû := jv j • φ j to the set J × W y (which indeed is an extension of u = j v j • φ j due tov j = v j andv i = 0 on φ −1 j ( τj 2 K) for i = j) and obtain a Hölder-continuous function, such that u =û| Wy∩Ω is also Hölderian on W y ∩ Ω with the same estimates.
Step 5. In order to deduce global Hölder continuity from the previous considerations, we need the following Lemma 3.25. There exists an ε > 0 such that, for every x ∈ Ω, the balls in Ω with center x and radius not larger than ε lie completely in at least one of the sets B xi or W y l .
Proof. Consider the function This function is continuous and strictly positive, since every y ∈ Ω is contained in at least one of the sets B xi or W y l . Therefore, it has to attain its minimum, say, ε > 0. Then it is straight forward to see that this ε fulfills the asserted condition, since at least one summand in the definition has to be bigger or equal to ε(y) for each y ∈ Ω. Now Lemma 3.25 in combination with Remark 2.8 iii) allows to fall back to the sets B xi and W y l ∩ Ω and thus implies global Hölder bounds on J × Ω, and this uniformly in f ∈ B step and in µ ∈ M n (κ 0 , κ 1 ).

Nonzero initial values and inhomogeneous Dirichlet boundary data.
Up to now, the fundamental difference between the approach in [40] and ours consists in the fact that here only the zero Dirichlet datum is allowed, which allowed to deduce global Hölder continuity for the solution (it is clear that also constant nonzero data is admissible by obvious modifications). In this chapter we will show a way how to admit (nonconstant) nonzero Dirichlet data -without losing the classical Hölder property for the solution. We restrict ourselves to the case where the Dirichlet datum does not depend on time. Moreover, aiming at Hölder continuity for the solution in both time and space, it is clear that the initial value must admit the correct boundary behaviour. In particular, in this context one can never expect that a solution with initial value 0 admits a nonzero Dirichlet datum.
We start with the introduction of the fundamental property for this chapter. Recall that we denote the (n − 1)-dimensional Hausdorff measure by H n−1 . Having this at hand, we can prove our first preparatory lemma. Proof. Consider for each z ∈ D\N the domain U z , the neighbourhoods U z,1 , . . . , U z,k of the connected components V z,1 , . . . , V z,k , the numbers τ z,j , and φ z,1 , . . . , φ z,k , the bi-Lipschitz mappings from Assumption 2.4 I). For z ∈ ∂D we collect the bi-Lipschitz mapping φ z and the neighbourhood U z of z from case II) of Assumption 2.4. For z ∈ D \ N we define another neighbourhood W z as follows: Letτ z ∈ ]0, τ z,1 [ be a number such that and define W z := φ −1 z,1 (τ z K) (this is well-defined since each U z,j is an open neighbourhood of z). Then the systems {U y } y∈∂D and {W z } z∈D\N form an open covering of D from which we choose a finite subcovering U y1 , . . . , U ym 2 , W z1 , . . . , W zm 3 , which allows to write D in the form D = m2 l=1 (U y l ∩ D) ∪ m3 l=1 (W z l ∩ D) . Thanks to the foregoing Remark 4.2, one has to show only that each of the sets U y l ∩D and W z l ∩D is a (n − 1)-set. For the sets D ∩ U y l this is immediate by Remark 4.2 ii) and the supposition on the mappings φ y l . For the sets W z l one has Let us now consider the terms ∂V z l ,j ∩ W z l , j = 1, . . . , k separately. From the definition of W z l it is clear that ∂V z l ,1 ∩ W z l is mapped by the bi-Lipschitzian transformation φ z l ,1 onto the setτ z l Σ. Thus, ∂V z l ,1 ∩ W z1 is a (n − 1)-set, thanks to Remark 4.2. This already assures the lower bound in (4.1) for the whole set D∩W z l . On the other hand, from the definition of W z l it follows that ∂V z l ,j ∩ W z l is mapped by the bi-Lipschitzian mapping φ z l ,j onto a subset of τ z l ,j Σ. Since τ z,j Σ admits the upper bound in (4.1), its subset φ z l ,j (∂V z l ,j ∩ W z l ) surely also does so. Finally, the upper bound for ∂V z l ,j ∩ W z l itself again follows from Remark 4.2. Hence, each of the sets U y l ∩ D and W z l ∩ D are a (n − 1)-set, making D also a (n − 1)-set.
Remark 4.5. Note that for q > n, W 1,q (Ω)-functions are Hölder continuous on Ω and thus in fact continuous up to the boundary of Ω, cf. Remark 2.8. So for ψ ∈ W 1,q (Ω), the pointwise restriction ψ| D is meaningful and indeed coincides with tr D ψ. We will use the notion ψ| D in the following.
Theorem 4.7. Adopt Assumption 2.4 and suppose q > n and s > 2(1 − n q ) −1 . Let ι ∈ B q,q 1− 1 q . Assume that u 0 ∈ W 1,q (Ω) satisfies u 0 | D = ι and let g be from L s (J; W −1,q D (Ω)). Then (4.3) admits exactly one solution w, and this solution is even Hölder continuous in space and time. The Hölder norm of w is uniformly bounded within the class µ ∈ M n (κ 0 , κ 1 ) for fixed g.
Remark 4.8. Following the strategy to split off the initial value requires u 0 to be in the domain of −∇ · µ(t, ·)∇ + 1 for each t ∈ J, which in general is only to be achieved if u 0 ∈ W 1,q (Ω), cf. Definition 2.9. Hence, in view of Proposition 4.4, the space B q,q 1− 1 q (D) for the boundary values on D is exactly the "optimal" one.

Global solvability of a non-linear heat equation and optimal regularity
for the solution. We show an application of the results established in the foregoing chapters. More specifically, we employ Theorem 2.13 to establish unique global existence of a solution to a quasilinear equation in divergence-form. We fix the following assumptions on the (nonlinear) forcing terms in the following problem: Remark 5.2. Assumption 5.1 is satisfied for a Carathéodory function F if the boundedness assumption holds true and for every R > 0 there exists a function L R ∈ L s (J) such that where w 1 , w 2 ∈ C(Ω) with w 1 C(Ω) , w 2 C(Ω) ≤ R.

A quasilinear heat-equation with optimal regularity for the solution.
Although we first have to introduce some auxiliary results for its proof (which, however, are of their own interest), this is the result: (Ω)) ∩ L s (J; W 1,q D (Ω)) of the quasilinear equation If F even satisfies the assumptions in Remark 5.2, this solution is unique.
Let us first compare Theorem 5.3 with other well-known general existence-and uniqueness theorems for quasilinear equations such as [43, Thm 3.1], which allow for more general data but yield only local solutions. The trade-off we make for global solutions, at this point, is twofold: First, we restrict ourselves to divergencetype operators, and secondly the requirements for the (nonlinear) inhomogeneity are stricter -we have to require uniform boundedness over C(Ω) and a slightly stronger Lipschitz condition. However, we emphasize that even for right-hand sides not depending on the function itself, e.g. [43,Thm. 3.1] does not yield global solutions, while our theorem/proof nearly immediately does, cf. Corollary 5.8. Moreover, we have the requirement of space dimension n = 3, whose necessity is a bit hidden: it is needed to guarantee uniformity of the domains of each of the operators −∇·φ(w)ρ∇ for varying w -which in turn is a common assumption -by using invariance under pertubation by continuous functions of the assumed isomorphism property of −∇ · ρ∇, which is only available for space dimension up to 3.
For the proof of Theorem 5.3 we, amongst others, need the following Lemma 5.4. Let ρ be a measurable coefficient function on Ω. Adopt Assumption 2.4 and assume that −∇ · ρ∇ is a topological isomorphism between W 1,q D (Ω) and W −1,q D (Ω) for some q > n.

D
(Ω), see [15,Thm. 6.2] -note that this is the (only) point in the proof where space dimension n = 3 is the limiting factor. In particular, the domain of the operators is uniformly W 1,q D (Ω). Since the mapping t → −∇ · ϕ(t)ρ∇ + 1 ∈ L(W 1,q D (Ω); W −1,q D (Ω)) (5.5) is continuous, [3,Thm. 7.1] shows existence of the unique solution u in the correct space. By [3,Thm. 3.1], this is equivalent to continuous invertibility of (∂ t + A(ϕρ), γ T0 ) −1 . Due to the operators A(ϕ k ρ) converge to A(ϕρ). This implies also convergence of (∂ t + A(ϕ k ρ), γ T0 ) −1 . Proof. We choose an arbitrary function u ∈ W 1,s (J; W −1,q D (Ω)) ∩ L s (J; W 1,q D (Ω)) with the initial value u(T 0 ) = w 0 (due to the very definition of the interpolation space which w 0 is from, this is always possible -we may, for instance, choose t → e ∇·ρ∇(t−T0) w 0 ). Note that, due to Lemma 5.4, u is a continuous function on J × Ω. Set w = u + v. The equation under consideration then becomes an equation in v, since u is fixed, that is, we now have to solve To this end, we consider for ψ ∈ C(J × Ω) the equation and define a function T (ψ) = v, such that v ∈ W 1,2 (J; W −1,2 D (Ω)) ∩ L 2 (J; W 1,2 D (Ω)) solves (5.7) (this is well-defined due to Proposition 2.10). Clearly, a fixed point of T would yield the searched-for solution for (5.6). Let us construct an appropriate setting: First, the set of all right-hand sides in (5.7) is bounded in the space L s (J; W −1,q D (Ω)) -boundedness of F in L s (J; W −1,q D (Ω)) over C(J × Ω) was an assumption -and for the divergence-term we estimate for every t ∈ J as follows: which is independent of ψ. As u is fixed, this means the right-hand sides in (5.7) are contained in a ball around the origin in L s (J; W −1,q D (Ω)), say, of radius r. Now set (Ω)) ≤ r, ζ ∈ C(J × Ω) and φ ≤ ζ ≤ φ as a subset of W 1,2 (J; W −1,2 D (Ω)) ∩ L 2 (J; W 1,2 D (Ω)). Theorem 2.13 shows that B is in fact contained in a ball Q α in some Hölder space C α (J × Ω), which in turn is compactly included in some ball Q c in C(J × Ω). Clearly, T maps Q c to B ⊂ Q α and the set {T (ψ) : ψ ∈ Q c } is compact in Q c . Hence, the Schauder fixed point theorem yields a fixed point v = T (v) in Q c , provided we are able to show continuity of the mapping T from Q c to Q c . So: The mapping ψ → φ(u+ψ) is continuous from Q c into C(J × Ω) by the Lipschitz assumption on φ, such that Lemma 5.5 implies that ψ → (∂ t + A(φ(u + ψ)ρ)) −1 is continuous from Q c to the linear bounded operators from L s (J; W −1,q D (Ω)) to W 1,s 0 (J; W −1,q D (Ω)) ∩ L s (J; W 1,q D (Ω)), cf. Remark 5.6. Thanks to the assumptions on F, ψ → F(·, u(·) + ψ(·)) is also a continuous map, hence the right-hand side R(ψ) in (5.7) depends continuously on ψ (here one also uses the Lipschitz property of φ). For a sequence ψ k → ψ in Q c we find via Lemma 5.4 T (ψ) − T (ψ k ) C(J×Ω) ≤ C (∂ t + A(φ(u(·) + ψ(·))ρ)) −1 R(ψ) −(∂ t + A(φ(u(·) + ψ k (·))ρ)) −1 R(ψ k ) W 1,s 0 (J;W −1,q D (Ω))∩L s (J;W 1,q D (Ω)) and a simple triangle argument shows that this goes to 0 as k goes to infinity since everything depends continuously on ψ. This is exactly the searched-for continuity of T . Finally, a fixed point v of T obviously solves (5.6) and is, thanks to Lemma 5.5, in fact from W 1,s 0 (J; W −1,q D (Ω)) ∩ L s (J; W 1,q D (Ω)), making w := u + v a solution of (5.1) in the optimal space W 1,s (J; W −1,q D (Ω)) ∩ L s (J; W 1,q D (Ω)). Concerning uniqueness, one observes that both the right-hand side F and the operator w → −∇ · φ(w)ρ∇w + w satisfy all assumptions in the theorem of Prüss [43,Thm. 3.1], if F satisfies the Lipschitz assumption in Remark 5.2. The quoted theorem then yields uniqueness of the solution.
Corollary 5.7. For fixed w 0 , consider the set of admissible data {ρ, φ, F} for the problem (5.1) as in the assumptions of Theorem 5.3, where κ 0 , κ 1 , φ, φ and C F are fixed. Then the set of associated solutions w ρ,φ,F is contained in a ball in some Hölder space C α (J × Ω).
Proof. Inspecting the proof of Theorem 5.3, one observes that the set B is always the same for all data {φ, F} when φ, φ and C F are fixed, and that the bound of B in the Hölder space is also uniform in κ 0 , κ 1 by Theorem 2.13. Hence, the size of the set Q α is also uniform in κ 0 , κ 1 , φ, φ and C F .
If the forcing term F in fact does not depend on w, we still obtain the following useful result from Theorem 5.3 and Corollary 5.7. In particular, the (nonlinear) solution operator, mapping f to w, transports bounded sets in L s (J; W −1,q D (Ω)) into bounded sets in C α (J × Ω) for some α > 0 and fixed w 0 . Remark 5.9. With Theorem 2.13 and essentially analogous techniques as displayed in this chapter, one might show existence of global Hölder-continuous solutions to semilinear equations with nonlinearities in the form as in Assumption 5.1, where the coefficient functions in the divergence-operator are only measurable in time. We omit the details.
6. Applications to optimal control. In this chapter we show that Hölder estimates, as established in various forms in the previous chapters, are not only interesting by their own right but may also put to good use in optimal control theory. The crucial point here is, of course, the compactness of bounded sets of Hölder functions in the space of continuous functions. We illustrate this in two ways, both of which translate weak convergence of the forcing terms induced by weak convergence of a parameter (or control ) to strong convergence of the associated solutions (or states) in the space of continuous functions. We do this for both a non-autonomous linear equation and a quasilinear equation as in Theorem 5.3. Applications for such properties in optimal control theory range from existence theory for optimal controls by standard arguments (see also Proposition 6.4 below) to second order sufficient conditions for optimal control problems, see e.g. [10] or [13]. An exemplary setting is given below.
Let X be a reflexive Banach space and consider a weakly continuous mapping E : X → L s (J; W −1,q D (Ω)). The first result is an immediate consequence of Theorem 2.13 by noting that the operators (∂ t + A(µ)) −1 are completely continuous from L s (J; W −1,q D (Ω)) to C(J × Ω). (Ω)) and u k ū in X . Then the solutions w k := w u k of converge strongly in C(J × Ω) to wū.
Remark 6.2. Proposition 6.1 may also be extended to nonconstant Dirichlet data and/or initial data from W 1,q (Ω) as in Ch. 4 in a straightforward way. We did not carry this out for the sake of simplicity.
Next, we add a control to the right-hand sides of the quasilinear problem in Theorem 5.3 in the following way: Let X be a reflexive Banach space such that X is a subset of the measurable functions on J with values in X. Let further (Ω) be such that i) for each u ∈ X , (t, w) → F(t, w, u(t)) satisfies Assumption 5.1 with the bound C F being uniform for u from bounded sets in X , ii) the mapping u → F(·, w(·), u(·)) is weakly continuous on X for each fixed w ∈ C(J × Ω). Note that the weak continuity assumptions on the superposition operators induced by E and F with respect to u are in particular satisfied if the dependence on u is affine-linear and continuous. Moreover, the assumptions imply that if w k →w in C(J × Ω) and u k ū in X , then F(·, w k (·), u k (·)) F(·,w(·),ū(·)) in L s (J; W −1,q D (Ω)). Theorem 6.3. Adopt the assumptions of Theorem 5.3 (the unique solutions case) and assume that the right-hand side is of the form F as above. Let u k ū be a weakly convergent sequence in X and w 0 ∈ (W 1,q D (Ω), W −1,q D (Ω)) 1 s ,s . Then the solutions w k := w u k of converge strongly in C(J × Ω) to wū.
Proof. Without loss of generality, we assume w 0 = 0 in the proof. One arrives at this situation by repeating the "split-off"-procedure done at the beginning of the proof of Theorem 5.3 and the obvious modifications from thereon without changing the fundamental properties of the problem, as seen there. The sequence (u k ) k is bounded in X . Due to the choice of s > 2(1 − n q ) −1 and Lemma 5.4, we have w k ∈ C(J × Ω) for each k. The assumptions on F and Corollary 5.7 then yield that the solutions w k are from a bounded set in C α (J × Ω) for some α > 0. Hence, there is a subsequence (w k l ) l of (w k ) k such that w k l →w in C(J × Ω). We need to show thatw = wū. Re-inserting the newly found convergence of w k l in the equations shows that the right-hand sides F(·, w k l (·), u k l (·)) now in fact converge weakly to F(·,w(·),ū(·)) in L s (J; W −1,q D (Ω)), while (∂ t + A(φ(w k l )ρ)) −1 goes to (∂ t + A(φ(w)ρ)) −1 by virtue of Lemma 5.5. However, the operators (∂ t +
We give an exemplary setting, where the previous result is needed: Suppose that we want to minimize a certain objective functional J : C(J × Ω) × X → R in such a way that w and u are connected by the equation w (t) − ∇ · φ(w(t))ρ∇w(t) + w(t) = F(t, w(t), u(t)), w(T 0 ) = w 0 , with w 0 ∈ (W 1,q D (Ω), W −1,q D (Ω)) 1 s ,s , where we suppose that the assumptions from Theorem 5.3 are satisfied such that for every u, (6.2) admits a unique solution w = w u which lies in W 1,s (J; W −1,q D (Ω)) ∩ L s (J; W 1,q D (Ω)). Moreover, let X ad ⊆ X be a subset of X representing admissible controls u, for instance such limited by upper and lower bounds on their function value (with ad like admissible). Then the problem under consideration is the quasilinear optimal control problem min w,u J (w, u) such that u ∈ X ad and (6.2) holds (QLOCP) with the set of feasible points C = (w, u) ∈ C(J × Ω) × X ad : (w, u) satisfy (6.2) .
A common problem arising in applications and modeled by (QLOCP) is that one wants to find a control u in such a way that the solution to the heat equation (6.2) at the end of a given process or simulation time described by the interval J matches a given temperature distribution w d as well as possible. In this case, the objective functional J may be chosen as where β > 0 is a regularization parameter and w d ∈ L 2 (Ω). The following theorem then shows that this task and generally a large class of optimal control problems is in fact well-posed in our setting: Proposition 6.4. Suppose the assumptions of Theorem 6.3 and assume that X ad is closed and convex. Let J be lower semi-continuous on C(J × Ω) × X and let u → J (w u , u) be coercive on X ad , if X ad is unbounded. Assume further that C is nonempty and that J is bounded from below over C. Then the problem (QLOCP) has an optimal solutionū ∈ X ad .
The reasoning is standard, see for instance [37, Ch. 1.5.2], but we give the proof for convenience of the reader: Proof. Due to the assumptions on C and J , we know that the there exists a minimizing sequence (w k , u k ) k ⊂ C such that lim k→∞ J (w k , u k ) = inf (w,u)∈C J (w, u).
If X ad is bounded or u → J (w u , u) is coercive there, the sequence (u k ) k is bounded in X and hence contains a weakly convergent subsequence u k l ū. The assumptions on X ad imply that it is weakly closed, henceū ∈ X ad . Theorem 6.3 further shows that w u k l → wū in C(J × Ω). Now the weakly lower-semicontinuity of J implies that lim l→∞ J (w u k l , u k l ) ≥ J (wū,ū), which means that J (wū,ū) = inf (w,u)∈C J (w, u).
7. Concluding remarks. It is not the intention of this paper to declare the concept of Ladyzhenskaja et. al in [40] to be outdated or not adequate any more. On the contrary, even nearly fifty years after it was first published, the results in [40] are still highly relevant -if not in their original form, then at least in a guiding and blue-print way, not accounting for the various hard facts it established. However, in view of the modern techniques for negative Sobolev spaces and Hölder spaces, an exposition of results in current, up-to-date mathematical "language" seems in order. In this sense, the preceding results could be seen as an adaption and translation of the classical results and deep insights in [40] to modern techniques.
We mention some open ends in the previous considerations: The results presented may be transferred to complex spaces as long as the coefficient functions in the equations are real. In this case, one may consider the realand imaginary parts in the considerations each on their own. Moreover, the "next step" in the great scheme would surely be maximal parabolic L p -regularity for non-autonomous equations with coeffients which are only measureable in time. While it is already known that maximal regularity for operators A(·) over an interval J implies maximal regularity for each of the autonomous operators A(s) for s ∈ J, up to now mostly some sort of continuity of the time-dependence is assumed additionally in order to conclude maximal regularity, see e.g. [3] for the corresponding result (already used in Lemma 5.5) and an overview. There is also a sequence of related, very recent work [5], [14], [30] and [42] which follows Lions' Theorem 2.10 (see [12,Ch. XVIII.3]) in a slightly different direction (maximal regularity over the Hilbert space H). Also very recent is a positive result on maximal L p regularity without any continuity assumptions on the time-dependence in [21]. Let us note that, in view of Ch. 6, maximal parabolic L p -regularity for only measurably time-dependent non-autonomous evolution equations would allow for a concise treatment of optimal control problems subject to these equations.
Finally, it would certainly be interesting to know which degree of Hölder continuity one obtains in Theorem 2.13 in dependence on the coercivity-constant κ 0 and upper bound κ 1 of the associated coefficient matrix. Based on [16,Ch. 4] for the elliptic case and the lack of related results apart from [40], at least such known to the authors, this seems like a difficult question which might be worth investigating.