Existence of variational solutions to a Cauchy–Dirichlet problem with time-dependent boundary data on metric measure spaces

The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal {X}}, d, \mu )$$\end{document}. For such parabolic minimizers that coincide with Cauchy-Dirichlet data η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} on the parabolic boundary of a space-time-cylinder Ω×(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \times (0, T)$$\end{document} with an open subset Ω⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \subset {\mathcal {X}}$$\end{document} and T>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T > 0$$\end{document}, we prove existence in the parabolic Newtonian space Lp(0,T;N1,p(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(0, T; {\mathcal {N}}^{1,p}(\varOmega ))$$\end{document}. In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.


Introduction
The aim of this paper is to show existence for parabolic minimizers on metric measure spaces. More precisely, we consider minimizers of integral functionals that are related to scalar functions u ∶ × (0, T) → ℝ which satisfy the inequality for a continuous integrand f ∶ [0, ∞] → ℝ . Here, ⊂ X is a bounded domain, where (X, d, ) is a metric measure space with a metric d and measure . 1 3 In the setting of a metric measure space, the classical calculus known from the Eucli dean space ℝ n is no longer available and instead of the gradient Du we have to introduce the notion of upper gradients. The "minimal upper gradient" will be denoted by g u .
The work at hand deals with parabolic minimizers on parabolic domains T ∶= × (0, T) with ⊂ X bounded and open and T > 0 . X denotes a metric meas ure space that fulfills a doubling property with respect to the metric d and the measure and supports a suitable Poincaré inequality. We refer to Chapter 2 for exact defini tions and the setting of the relevant spaces. Here, we are going to generalize results which have recently been proven in [13], while we follow the methods in [7]. There, the authors have considered an obstacle problem in the Euclidean case.
Since the beginning of the twentyfirst century, doubling measure spaces have been studied quite extensively, see for example [10,18,25,26,28,30,39,53,54] and espe cially [3] for an overview and further references. The idea of considering variational problems on metric measure spaces is based on independent proofs of Grigor'yan [23] and SaloffCoste [51] of the fact that on Riemannian manifolds the doubling property and Poincaré inequality are equivalent to a certain Harnacktype inequality for solutions of the heat equation. But instead of Riemannian manifolds, we are interested in more general spaces.
Existence for parabolic problems on metric measure spaces has already been dealt with in [13] with boundary data independent from time. This paper deals with time dependent boundary data and therefore uses a different approach. In the elliptic case, the Dirichlet problem for pharmonic functions (i.e. f ( ) = p ) has been considered by Björn and Björn in [3, Chapters 8 and 10] and [4], the latter a joint work with Shanmu galingam. By the results of Cheeger in [10], where it is shown that his definition of Sobolev spaces on metric spaces leads to a reflexive space via the existence of a differ ential as a measurable section of a finite dimensional cotangent bundle, one can see that direct methods in the calculus of variations can be applied to prove the existence for the pDirichlet problem, see also [54]. The investigation of parabolic problems on metric measure spaces started not long ago with the work of Kinnunen, Marola, Miranda and Paronetto, see [35], concerning regularity problems. Since then, the most contributions in this field of research have been made to stability theory (see e.g. [20,21,43]) and regularity problems (see also [19,24,[46][47][48]). When concerning issues of regularity, one tries to establish for instance Hölder continuity of a solution that is assumed to be an element of the parabolic Newtonian space L p (0, T;N 1,p ( )) . By that function space, we denote those u ∶ (0, T) → N 1,p ( ) , such that the mapping t ↦ ||u(t)|| p N 1,p ( ) is integra ble over the interval (0, T).
Our aim is to show existence for a parabolic minimizer of a functional in such a parabolic Newtonian space, which will be done via the concept of global vari ational solutions. The integrand is required to be convex and to fulfill a pgrowth assump tion of the form for all ∈ [0, ∞] with constants , L > 0 , 1 < p < ∞ . Also, we are going to assume f to be increasing. Again, we refer the reader to Chapter 2 for the exact definitions and in particu lar to Theorem 2.3 to find the exact statements.
In the paper at hand, we are going to put the focus on how to overcome certain dif ficulties given by the setting of metric measure spaces. The main difficulty that leads to such obstacles is given by the fact that the upper gradient g u unlike the classical gradient Du does not behave linearly but only in a sublinear way, such as the modulus |Du| does. In fact, for X = ℝ n there holds g u = |Du| . Therefore, if we are going to generalize results that have been found for integrands f ∶ ℝ n → ℝ in the Euclidean case, we have to think about what happens if the integrand only depends on the modulus, i.e. f ( ) = g(| |) for a convex function g. Such integrands have been looked at before in the Euclidean case, see i.e. [11,12,17,22,44,45]. Since these papers are dealing with regularity issues, additional assumptions had to be made there, so are the authors in [22] for example looking at integrands F(|Du|) such that F and F(t) t p are increasing and F(t) t q is decreasing for p < q . In [45] the stationary elliptic case is considered for vectorvalued problems, see also [55]. Now for the existence problem, if we require f ∶ ℝ n → ℝ to be convex, a monotonic ity assumption is in order. This assumption (that has also been made in the papers quoted before) comes in a natural way as we can easily see as follows: for ∈ ℝ n and ∈ [0, 1] , the convexity of g shows i.e. by letting s ∶= (1 − 2 ) ∈ [−1, 1] we have Now let 1 , 2 ∈ ℝ n with 0 < | 1 | ≤ | 2 | . Then, for ∶= 2 and s ∶= Hence also in our case, the monotonicity assumption on f ∶ [0, ∞] → ℝ does not impose a loss of generality. These assumptions cover all well known examples, such as the penergy, Orlicztype integrands and integrands without polynomial structure at all, such as: Here, 1 < p < ∞ and r ≥ 1 are arbitrary integrability exponents.
Our method of proof is aligned to the methods in the work of Bögelein, Duzaar and Scheven [7]. For a fixed step size h ∈ (0, 1] we will inductively construct a sequence u i of minimizers to elliptic variation functionals for timediscretized boundary values corresponding to the given boundary data in the Cauchy-Dirichlet problem we are considering. Then, for t ∈ ((i − 1)h, ih] we let and via certain energy estimates we can show that the limit map u in the sense of weak starconvergence as h ↘ 0 fulfills the variational inequality for any admissible comparison function v ∶ T → ℝ that coincides with on the lateral boundary × (0, T) , where 0 denotes the initial datum. This is called the technique of minimizing movements, see also [1,15] for further references. To show the energy esti mates mentioned before we apply a Jensentype inequality established in [49]. Finally, to obtain the existence result for more regular data, we introduce a mollification in time.

Notations
Let (X, d, ) be a separable, connected metric measure space, i.e. (X, d) is a complete, sepa rable and connected metric space endowed with a Borel measure on X . The measure is assumed to fulfill a doubling property, i.e. there exists a constant c ≥ 1 , such that for all radii r > 0 and centres x ∈ X . Here B r (x) ∶= {y ∈ X ∶ d(x, y) < r} denotes the open ball with radius r and centre x with respect to the metric d. The doubling constant is defined as A complete metric measure space that fulfills the doubling property is proper, meaning that all closed and bounded subsets are compact, see [3,Proposition 3.1].
Following the concept of Cheeger [10], Heinonen and Koskela [30], we call a Borelfunction g ∶ X → [0, ∞] an upper gradient for an extended realvalued func tion u ∶ X → [−∞, ∞] if for all x, y ∈ X and rectifiable curves ∶ [0, L ] → X with (0) = x, (L ) = y there holds Since this definition clearly depends on the metric d and the measure , we abuse the notation N 1,p ( ) as an abbrevation for N 1,p ( , d, ) . For more details on metric measure spaces we refer the reader to [3]. As in [3], we define the Sobolev pcapacity of an arbitrary set E ⊂ X with respect to the space N 1,p (X) as where the infimum is taken over all those functions u ∈ N 1,p (X) such that u| E ≥ 1 holds almost everywhere (in short a.e.). A property is said to hold pquasi everywhere (in short pq.e.) if the set for which the property fails is of pcapacity zero.
We follow the method of [34] and define for an arbitrary set E ⊂ X the space Ñ 1,p 0 (E) as the set of all functions u ∶ E → [−∞, ∞] for which there exists a function ũ ∈Ñ 1,p (X) that fulfills ũ = u almost everywhere on E and ũ ≡ 0 pquasi everywhere on X⧵E . By setting u ∼ v if and only if u = v almost everywhere on E we establish an equivalence relation on Ñ 1,p 0 (E) and define as the Newtonian space with zero boundary values, equipped with the norm

Parabolic Newtonian spaces
Since we are looking at problems with timedependence, we have to give a definition for the parabolic Newtonian space L p (0, T;N 1,p ( )) and its local version, respectively. For T > 0 , we denote the spacetime cylinder over an open subset ⊂ X as T ∶= × (0, T) . The parabolic space consists of all functions u ∶ T → ℝ for which the mapping is strongly measurable and the function (0, Here, strongly measurable means measurable in the sense of Bochner, i.e. there exists a sequence of simple functions k ∶ (0, is endowed with the norm Remark 2.1 Since we are dealing with timedependent problems, the functions u we are concerned with are functions u ≡ u(x, t) depending on the variable x in the metric space X and on the time variable t in ℝ . Therefore we have to understand the concept of weak upper gradients in the sense of time slices, i.e. for a function u ∈ L p (0, T;N 1,p ( )) the parabolic minimal pweak upper gradient of u is defined as surable since the spaces L p (0, T;L p ( )) and L p ( T ) coincide, see also [13]. This allows us to apply Fubini's theorem to parabolic upper gradients. Now, following [3, Chapter 2.6], we define the space L p loc (X) as the set of all func tions u ∶ X → ℝ , such that for every x ∈ X there exists r x > 0 with u ∈ L p (B r x (x)) . Just like that, we can define N for all 0 < t 1 < t 2 < T and ′ ⋐ . If clear from the context, we will abbreviate For a Banach space B and T > 0 , the space Naturally, for ∈ (0, 1] , the space consists of those functions u ∈ C 0 ([0, T];B) , for which additionally holds true.

Poincaré inequality
In addition to the doubling property, we demand that the metric measure space (X, d, ) supports a weak (1, p)Poincaré inequality, in the sense that there exist a constant c P > 0 and a dilatation factor ≥ 1 such that for all open balls B (x 0 ) ⊂ B (x 0 ) ⊂ X , for all L p functions u on X and all upper gradients g u of u there holds where the symbol denotes the mean value integral of the function u on the ball B (x 0 ) with respect to the measure . We will omit the word 'weak' from here on and simply refer to this inequality as 'Poincaré inequality'. Poincaré inequalities on metric measure spaces have been studied quite extensively in the literature, see for example [2,5,31,33,36,38,40,51,52].

Variational solutions
In the Euclidean case, i.e. X = ℝ n , one might consider the CauchyDirichlet problem where par T ∶= ( × {0}) ∪ ( × (0, T)) denotes the lateral boundary of T . By an idea of Lichnewsky and Temam [42], one can define the concept of variational solutions to CauchyDirichlet problems like (2.6). Since on a metric measure space we do not have the possibility to explain derivatives such as in (2.6), we cannot consider CauchyDirichlet problems like this. But the concept of variational solutions is described on a purely vari ational level, so it can be extended to the concept of metric measure spaces. The formu lation of the problem takes place in the parabolic function space + L p (0, T;N 1,p 0 ( )) , consisting of those v ∈ L p (0, T;N 1,p ( )) , such that for almost every time slice t ∈ [0, T] [32,50] for details on L p (0, T;N 1,p ( )) . We consider boundary data ∶ T → ℝ for which there holds We note that by assumption (2.7) the map is an admissible comparison function in the variational inequality (2.10). This allows the deduction of certain energy bounds and shows that variational solutions attend the initial datum 0 in the L 2 sense, see below.
of the Cauchy-Dirichlet problem in the sense of Definition 2.2. Furthermore, the solution admits the initial datum 0 in the L 2 sense.

Properties of metric measure spaces
A direct consequence of the doubling property (2.1) is that for every ball Moreover, since X is connected there exists a constant c > 0 and an exponent > 0 , such that The number n d in (3.1) plays the role of a 'dimension from below', related to the measure ; but since c d in general isn't of the form 2 m for an integer m, we point out that n d cannot expected to be an integer either.
Since X is a doubling space, it supports the Vitali covering theorem (see [29]) and therefore also the differentiation theorem of Lebesgue, i.e. for every nonnegative locally integrable function on X we have that for almost every x ∈ X . Now we collect some calculus rules for upper gradients in met ric measure spaces. For the proofs, we refer the reader to [3].

Remark 3.2
Even if we choose the minimal pweak upper gradients g u and g v , respectively, the function |u|g v + |v|g u is in general not the minimal pweak upper gradient for uv. Coun terexamples can be found in the book [3]. This, of course, also applies to the sum u + v , which can easily be seen by taking v = −u.
Another useful result is the following: Lemma 3.3 (Theorem 2.18 in [3]) Let u, v ∈ N 1,p ( ) and be Lipschitz on , such that is a pweak upper gradient of w.

Remark 3.4
All these rules can be understood also for timedependent functions u ∈ L p (0, T;N 1,p ( )) , as we can define upper gradients for those functions as in (2.4).
Another significant property of minimal pweak upper gradients is that they are local in the sense that two functions coincide on a set, then also their minimal upper gradients coin cide on this set. The result is taken from [3, Chapter 2.4].

Poincaré inequalities and Sobolev embedding
It follows directly from Hölder's inequality that if a metric measure space supports a (1, p)Poincaré inequality, then it supports a (1, q)Poincaré inequality for all q ≥ p . On the other hand, it was shown in [33] that if a complete metric space is endowed with a dou bling measure and supports a (1, p)Poincaré inequality, then it also supports a (1, p − ) Poincaré inequality for some > 0 (and therefore also a (1, q)Poincaré inequality for all q ≥ p − ). From [28] we know that if we assume a weak (1, p)Poincaré inequality, then a Sobolev type embedding theorem holds and therefore the space supports a weak (q, p) Poincaré inequality for all q ≤ p * , with In [37], see also [27,28,30], it was shown that in this case for every u in the space N 1,p (B 2 (x 0 )) with B 2 (x 0 ) ⊂ X the following Sobolevtype inequality holds: with a constant c * ≡ c * (c d , c P ) > 0 . Poincaré and Sobolev inequalities also hold on more general domains. In detail, the Poincaré inequality holds on bounded measurable subsets E of the metric space X such that the pcapacity of the complement X⧵E is positive, i.e. we have for every function u ∈ N 1,p 0 (E) and for every bounded measurable set E ⊂ X with Cap p (X⧵E) > 0 with a constant C E ≡ C E (c P , c d , p, E) > 0.

Smoothing procedures in time
Since variational solutions in the sense of Definition 2.2 are not regular with respect to time, they are in general not admissible as comparison maps in (2.10). To overcome this difficulty, we are going to construct test functions with the correct regularity with respect to time by fol lowing an approach introduced by Landes [41]. The idea is to construct the mollification For maps v ∈ L r (0, T;B) with t v ∈ L r (0, T;B) we have the following assertion. T;B) . Then, for the mollification in time defined by the time derivative can be computed by and moreover, we have that

Miscellaneous
The following results concerning the initial condition u(0) = 0 , energy estimates, con tinuity in time and uniqueness can be shown analogously to the Euclidean case, see [7]. Note that for our purposes Lemmas 3.6, 3.7 and the following technical lemma that can be proven similarly to [ (2.7). Then, every variational solution of the Cauchy-Dirichlet problem in the sense of Definition 2.2 satisfies u ∈ C 0 ([0, T];L 2 ( )).

Lemma 3.12 Assume that fulfills (2.7). Then, there is at most one variational solution
of the Cauchy-Dirichlet problem with initial and boundary data in the sense of Definition 2.2.
Note that for the proof of Lemma 3.12, the functions we use as comparison maps in (2.10) are defined as v h ∶= [v − ] 0 h + , similarly as in Lemma (3.8). Furthermore, it can be shown that a variational solution u to the CauchyDirichlet problem in the sense of Definition 2.2 on the cylinder T is also a variational solution on subcylinders for any ∈ (0, T) . The proof is also analagous to the Euclidean version, see [7].

Existence for regular data
In this section we are going to treat the case of sufficiently regular boundary data. For the proof of Theorem 4.1 we shall proceed in several steps.

A sequence of minimizers to elliptic variational functionals
We fix a step size h ∈ (0, 1] and write i ∶= (ih) ∈ L 2 ( ) ∩ N 1,p ( ) for each i ∈ ℕ 0 with ih ≤ T for the timediscretized boundary value as in [7]. Our goal will be to inductively construct a sequence u i ∈ L 2 ( ) ∩ ( i + N 1,p 0 ( )) of minimizers to cer tain elliptic variational functionals. The precise construction is as follows. Suppose that u i−1 ∈ L 2 ( ) ∩ ( i−1 + N 1,p 0 ( )) for some i ∈ ℕ has already been defined.
shall be the initial boundary datum. Then, we let u i be the minimizer of the variational functional in the class of functions v ∈ L 2 ( ) ∩ ( i + N 1,p 0 ( )) . Note that this class is nonempty since v = i is admissible. The existence of u i can be deduced by means of standard compactness arguments using the convexity of f combined with the monotonicity condition (2.8) 2 . More detailled, let and (v n ) n∈ℕ ⊂ K i be a minizing sequence of i . By making use of the Poincaré inequality (3.2) we can show that there holds . Thus, there is a subsequence of (v n ) that converges weakly to some v in K i as n → ∞ . By applying Mazur's lemma, we find a sequence (w m ) m∈ℕ of convex combinations of the v n , i.e.
such that w m ⟶ v strongly in K i as m → ∞ , i.e.
By the convexity of i , the w m are also a minimizing sequence of i in the class K i . By the Lipschitz property (2.9) of f we can deduce lower semicontinuity of i with respect to strong convergence in K i , which shows that v is a minimizer of i in the class K i . By the strict convexity of the second term in i , we obtain uniqueness.

Energy estimates
We first observe that u i−1 + i − i−1 is an admissible comparison function in the functional i for any i ∈ ℕ . Therefore, by using the minimality of u i we find that For the integral we have that the estimate holds true by means of Jensen's inequality. Using in turn the Lipschitz estimate (2.9) for f, Young's inequality, the assumption h ≤ 1 and the growth condition (2.8), we deduce for the integrand of the following bound from above: with a constant c ≡ c(p, L, ) . Note that there holds From [49, Theorem 4.1] we therefore obtain the Jensentype inequality due to the fact that the mapping u ↦ g u is convex and Wcontinuous, where Inserting this into we obtain for a constant c ≡ c(p, L, ) . Together with the estimates for the integral this leads to the following preliminary energy estimates Summing up (4.3) from i = 1, ..., l for some l ∈ ℕ with lh ≤ T , we conclude that Here we used the shorthand notation for ∈ (0, T] . Reabsorbing in the preceding inequality the term on the lefthand side, we obtain from the last inequality that Next, we iterate inequality (4.3) from j = 1, ..., i with i ∈ {1, ..., l − 1} . This leads to Inserting this into the penultimate inequality yields Noting that (1 + h) l ≤ (1 + T l ) l ≤ e T , the preceding inequality leads to the bound for any l ∈ ℕ with lh ≤ T . From (4.4) we also conclude that for any l ∈ ℕ with lh ≤ T.

The limit map
From now on, we consider only such values h ∈ (0, 1] for which l ∶= T h ∈ ℕ holds true. Then, we define a function u (h) ∶ × (−h, T] → ℝ by From (4.4), (4.5) and the growth condition (2.8) on f we know that holds true. The energy estimate (4.7) ensures that the sequence u (h) is bounded in L ∞ (0, T;L 2 ( )) and L ∞ (0, T;N 1,p ) . Moreover, we have that there holds u (h) ∈ (h) + L ∞ (0, T;N 1,p 0 ( )) . At this point, some remark on the latter function space is in order. The space L ∞ (0, T;N 1,p ( )) is isometrically isomorphic to the dual space of L 1 (0, T;(N 1,p ( )) * ) , where by (N 1,p ( )) * we denote the dual space of N 1,p ( ) . This holds since the space N 1,p ( ) is reflexive as a closed subspace of the reflexive Banach space L p ( ) . See [16, Chapter 4, Section 1] for more details on dual spaces of Bochner spaces. Since N 1,p ( ) is also separable for 1 ≤ p < ∞ , so is the space L 1 (0, T;(N 1,p ( )) * ) . There fore, there exists a subsequence (h k ) k∈ℕ with h k ↘ 0 as k → ∞ and a limit function Together with (4.4) and the convexity of f this implies Therefore, also the family of ũ (h) is bounded in L ∞ (0, T;N 1,p ( )) and the family of tũ (h) is bounded in L 2 ( T ) . Hence, there exists a subsequence (h k ) k∈ℕ with h k ↘ 0 as k → ∞ and a limit function ũ ∈ L ∞ (0, T;N 1,p ( )) with tũ ∈ L 2 ( T ) such that holds in the limit k → ∞ . The strong convergence in L min{2,p} ( T ) is secured by a met ric version of the RellichKondrachov theorem, see [43,Theorem 4.1]. This theorem can be applied since we assumed the doubling condition (2.1) and the weak (1, p)Poincaré inequality (2.5). Since we have we conclude from (4.4) that which by means of Hölder's inequality implies Joining this with (4.10) 1 implies that also u (h k ) →ũ strongly in L min{2,p} ( T ) as k → ∞ and hence ũ = u . By weak lower semicontinuity and (4.9) we infer that (4.10) By definition of H , this yields the claimed energy estimate (4.1).

Minimizing property of the approximations
Now, we observe that u (h) is a minimizer of the functional in the class of functions v ∈ L 2 ( T ) ∩ (h) + L p (0, T;N 1,p 0 ( )) . Indeed, for any admissi ble function v the following computation shows that holds true. Here we used the definition of u (h) from (4.6), the minimizing property of u i and the very definition of the functional (h) . Following the strategy of [7], this leads to the estimate

The variational inequality for the limit map
Here we pass to the limit h ↘ 0 in (4.11) as was done in [7]. In order to do that we need to replace the boundary condition v = (h) on the lateral boundary by the hindependent con dition that v = . Therefore, we consider a comparison map v ∈ + L p (0, T;N 1,p 0 ( )) with (4.11) 1,p 0 ( )) . Since g (h) − → 0 strongly in L p ( ) as h ↘ 0 , which can be deduced by using the Jensentype inequality (4.2), we infer by means of the Lipschitz condition (2.9) on f that holds true in the limit h ↘ 0 , i.e. we have In order to treat the other terms appearing in (4.11) we follow again the methods in [7] to find that after passing to the limit h ↘ 0 there holds for any comparison function v ∈ + L p (0, T;N 1,p 0 ( )) with t v ∈ L 2 ( T ) and v(0) ∈ L 2 ( ) . This means that u is a variational solution to the CauchyDirichlet problem, which con cludes the proof of Theorem 4.1. ◻

Approximation by regular data
In this section we are going to follow [7] and establish the existence result for general boundary data with The proof is divided in several steps.

Regularization
We consider a sequence (h i ) i∈ℕ with h i > 0 ( i ∈ ℕ ) and h i ↘ 0 as i → ∞ and let i ∶= h i holds true for all i ∈ ℕ . Then, we define mollifications in time by is defined as in (3.4). By Lemma 3.6 we have i ∈ L 2 ( T ) ∩ L p (0, T;N 1,p ( )) and according to (3.3) there also holds For the comparison maps for the limit problem we use the same mollification as for the boundary values , i.e. for v ∈ + L p (0, T;N . By definition, the mollifications satisfy v i ∈ i + L p (0, T;N 1,p 0 ( )) . According to Lemma 3.6 we have that Next, we claim for all i ∈ ℕ that which implies in particular that the sequence ( t v i ) i∈ℕ is bounded in the space L 1 (0, T;L 2 ( )) . For the proof of this claim we define ṽ i ∶= v i + e − t h i (v(0) − (i) 0 ) and note that Lemma 3.7 together with Jensen's inequality implies the bound Moreover, we have that holds true for all i ∈ ℕ . Joining the two preceding estimates, we find that the assertion (4.15) holds true. Our next goal is the proof of the convergence To this end, we consider

Solutions of the regularized problem
As a consequence of (4.14), the boundary data i and the initial datum 0 satisfy all assump tions from the preceding steps, i.e. from Sect. 4.1, Theorem 4.1. Hence, we obtain solutions of the variational inequalities for a.e. ∈ [0, T] and every comparison map v ∈ i + L p (0, T;N 1,p 0 ( )) with t v ∈ L 2 ( T ) and v(0) ∈ L 2 ( ) . In particular, we can apply Lemma 3.10 to the variational solutions u i and the comparison function v = i in order to deduce the estimate The first two terms on the righthand side stay bounded in the limit i → ∞ as a conse quence of (4.16) and (4.17) applied to v = and the growth assumption (2.8) 1 on f. The last term in (4.21) stays bounded according to Lemma 3.6 with r = ∞ and B = L 2 ( ) . Hence, the estimate (4.21) together with the coercivity (2.8) 1 of f implies that the sequence (u i ) i∈ℕ is bounded in the spaces L ∞ (0, T;L 2 ( )) and L p (0, T;N 1,p ( )).