The bounded slope condition for parabolic equations with time-dependent integrands

In this paper, we study the Cauchy–Dirichlet problem ∂tu-divDξf(t,Du)=0inΩT,u=uoon∂PΩT,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {\text {div}} \left( D_\xi f(t, Du)\right) = 0 &{} \quad \hbox {in} \ \Omega _T, \\ u = u_o &{} \quad \hbox { on} \ \partial _{\mathcal {P}} \Omega _T,\\ \end{array} \right. \end{aligned}$$\end{document}where Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^n$$\end{document} is a convex and bounded domain, f:[0,T]×Rn→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:[0,T]\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}$$\end{document} is L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-integrable in time and convex in the second variable. Assuming that the initial and boundary datum uo:Ω¯→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_o:{\overline{\Omega }}\rightarrow {\mathbb {R}}$$\end{document} satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.

Surprisingly, while Hardt and Zhou [16,Chapter 4] used the bounded slope condition in a regularity argument in a time-dependent setting involving functionals with linear growth, an evolutionary analogue of the above stationary theorem was established only rather recently by Bögelein, Duzaar, Marcellini and Signoriello [7].They considered the Cauchy-Dirichlet problem L. Schätzler and J. Siltakoski where Ω T := Ω × (0, T ) with Ω ⊂ R n and T ∈ (0, ∞] denotes a space-time cylinder and ∂ P Ω T := ∂Ω × (0, T ) ∪ (Ω × {0}) its parabolic boundary.Given a Lipschitz continuous initial and boundary datum u o that satisfies the bounded slope condition, in [7] it was proven that the above problem admits a unique variational solution that is globally Lipschitz continuous with respect to the spatial variables.Moreover, if the integrand f fulfills an additional p-coercivity condition with some p > 1, Bögelein and Stanin [8] obtained the local Lipschitz continuity of variational solutions in space and time under the assumption that u o is convex and Lipschitz continuous.Further, global continuity of u was proven in the case that Ω is uniformly convex.
For the same class of integrands and merely convex domains Ω, Stanin [30] showed that variational solutions are still globally Hölder continuous even if the convexity assumption on u o is dropped.Equations with lower-order terms were considered by Rainer, Siltakoski and Stanin [27] who extended a stationary Haar-Rado type theorem by Mariconda and Treu [24] to the parabolic problem where f is convex and p-coercive with some p > 1 and the lower-order term g satisfies a technical condition, in particular convexity with respect to u.As a corollary, the authors in [27] obtained the global Lipschitz continuity with respect to the spatial variables of variational solutions under the classical twosided bounded slope condition provided that f ∈ C 2 is uniformly convex in a suitable sense.
Existence and regularity of solutions under general growth conditions, such as the so called p − q-growth conditions, have been recently considered by many authors, see for example [21,25] and the references therein.We emphasize that in the present manuscript, because of the bounded slope condition, no special growth conditions are imposed on the elliptic part of the operator.
The objective of the present paper is to extend the result of [7] to include time-dependent integrands.In order to focus on the novelty and to include integrands f with linear growth, we consider the classical bounded slope condition and avoid lower-order terms.We are concerned with parabolic partial differential equations of the form where Ω ⊂ R n is a convex and bounded domain and T ∈ (0, ∞].The integrand f : [0, T ] × R n → R is assumed to be a Carathéodory function that satisfies the following assumptions: t → f (t, ξ) ∈ L 1 (0, τ) for all ξ ∈ R n and τ ∈ (0, T ] ∩ R. (1.2) In particular, for any L > 0 and τ ∈ (0, T ] ∩ R the map t → max |ξ|≤L |f (t, ξ)| belongs to L 1 (0, τ) (see Sect. 2.3 below).Therefore, for any τ ∈ (0, T ] ∩ R and V ∈ L ∞ (Ω T , R n ) we have that

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The bounded slope condition for parabolic equations Page 3 of 34 76 We emphasize that t → f (t, ξ) is neither assumed to be continuous nor weakly differentiable.
In the present paper, we define variational solutions in the same way as in [5].This notion of solution, inspired by Lichnewsky and Temam [20], was introduced by Bousquet [2,3] in the time-independent setting.We consider the following class of functions that are Lipschitz continuous in space Further, we denote the subclass related to time-independent boundary values 2) and consider a boundary datum u o ∈ W 1,∞ (Ω).In the case T ∈ (0, ∞) a map u ∈ K ∞ uo (Ω T ) is called a variational solution to the Cauchy-Dirichlet problem associated with (1.1) and u o in Ω T if and only if the variational inequality ) is a variational solution in Ω τ for any τ ∈ (0, ∞), u is called a global variational solution or variational solution in Ω ∞ to the Cauchy-Dirichlet problem associated with (1.1) and u o .
Our main result concerning the existence of variational solutions which are Lipschitz continuous with respect to the spatial variables can be formulated as follows.

Theorem 1.2. Let Ω ⊂ R n be an open, bounded and convex set and
(1.4) L. Schätzler and J. Siltakoski NoDEA Furthermore, we show that variational solutions to (1.1) are weak solutions and consequently, they are 1/2-Hölder continuous in time provided that the map ξ → f (t, ξ) is C 1 and uniformly locally Lipschitz in the following sense: For each L > 0, there exists a constant M L > 0 such that sup t∈(0,T ) (1.5) Theorem 1.3.Suppose that the assumptions of Theorem 1.2 hold.Moreover, assume that the mapping ξ → f (t, ξ) is in C 1 (R n ) for almost all t ∈ (0, T ) and satisfies (1.5).Then the unique variational solution u to the Cauchy-Dirichlet problem associated with (1.1) and u o is a weak solution (see (7.1)).Further, it is contained in the space of Hölder continuous functions C 0;1,1/2 (Ω T ).
To prove Theorem 1.2, we may assume without a loss of generality that T < ∞, see the beginning of Sect.6.The proof is divided into three parts.We first assume that the integrand is suitably regular and in particular has a weak derivative with respect to the time variable.Then the method of minimizing movements yields a solution u to the so called gradient constrained obstacle problem to (1.1), where the L ∞ -norms of the gradients of the solution and the comparison maps are bounded by a fixed constant L ∈ (0, ∞).Moreover, the regularity assumption on f ensures that u has a weak time derivative in L 2 (Ω T ).
Next, under the same regularity assumptions on f as in the first step, a standard argument exploiting the bounded slope condition and the maximum principle yields the uniform gradient bound (1.4) for u.Choosing L large enough, this in turn allows us to deduce that u is in fact already a solution to the unconstrained problem in the sense of Definition 1.1.
To deal with a general integrand f , we consider its Steklov average f ε .Since f ε admits a weak time derivative, by the results mentioned in the preceding paragraph there exists a solution u ε to the Cauchy-Dirichlet problem associated with f ε in the sense of Definition 1.1.Moreover, since for each ε > 0 the solution u ε satisfies the gradient bound (1.4) and u ε = u o on ∂Ω × (0, T ), there exists a limit map u ∈ L ∞ (Ω T ) such that u ε → u uniformly and Du ε * Du weakly * up to a subsequence as ε ↓ 0. This allows us to conclude that u is a variational solution in the sense of Definition 1.1, finishing the proof of Theorem 1.2.The proof of Theorem 1.3 is similar to the one found in [7,Chapter 8].The C 1 assumption on the integrand ensures the validity of the weak Euler-Lagrange equation, which lets us apply the argument from [6, pp. 23-24] to prove a Poincaré inequality for variational solutions.The Hölder continuity then follows from the Campanato space characterization of Hölder continuity by Da Prato [9].
The paper is organized as follows.Section 2 contains preliminary definitions and basic observations about the integrand.In Sect. 3 we prove certain properties of variational solutions that are required in later sections, including the comparison and maximum principles.Under additional regularity assumptions on f we use the method of minimizing movements to prove the existence of variational solutions to the gradient constrained problem in Sect. 4 and in Sect. 5 we consider the unconstrained problem.Finally, in Sect.6 we consider general integrands and finish the proof of Theorem 1.2 and Hölder continuity in time is proven in Sect.7 under additional regularity assumptions.

Notation
Throughout the paper, for p ∈ [1, ∞] and m ∈ N the space L p (Ω, R m ) denotes the usual Lebesgue space (we omit R m if m = 1) and W 1,p (Ω) and W 1,p 0 (Ω) denote the usual Sobolev spaces.If Ω is a bounded Lipschitz domain, W 1,∞ (Ω) can be identified with the space up to the boundary of Ω.Note that in particular any convex set has a Lipschitz continuous boundary, since convex functions are locally Lipschitz [11,Corollary 2.4].Further, for a Banach space X and an integrability exponent p ∈ [1, ∞] we write L p (0, T ; X) for the space of Bochner measurable functions v : [0, T ] → X with t → v(t) X ∈ L p (0, T ).Moreover, C 0 ([0, T ]; X) is defined as the space of the continuous functions v : [0, T ] → X.For maps v defined in Ω T we also use the short notation v(t) for the partial map x → v(x, t) defined in Ω.Finally, for a set A ⊂ R m , the characteristic function χ A : R m → {0, 1} is given by χ A (x) = 1 if x ∈ A and χ A (x) = 0 else.

Bounded slope condition
In the proof of the existence result in Sect. 5 ).Note that unless U itself is affine, the convexity of Ω is necessary for the bounded slope condition to hold.Even strict convexity of Ω is not sufficient for general U , since the boundary can become "too flat".However, we know that for a uniformly convex, bounded C 2 -domain Ω and v ∈ C 2 (R n ) the restriction U = v| ∂Ω fulfills the bounded slope condition.For more details, we refer to [14,26].On the other hand, in the parabolic setting the following example is relevant: Consider a convex domain Ω with flat parts (such as a rectangle) and a Lipschitz continuous function u o that vanishes at the boundary of Ω; i.e. we prescribe zero lateral boundary values, but the initial datum is not necessarily identical to zero.
We need the following lemma from [

Dominating functions for the integrand
Observe that for any L > 0 the map t → max |ξ|≤L f (t, ξ) is measurable, since we have that max |ξ|≤L f (t, ξ) = max ξ∈BL(0)∩Q n f (t, ξ) and the maximum of countably many measurable functions is measurable.The same holds true for t → min |ξ|≤L f (t, ξ).In the following lemma, we show that they are contained in L 1 (0, T ).

Lower semicontinuity
In the course of the paper we will need the following result on the lower semicontinuity of integrals involving f with respect to the weak * topology of Then, for any sequence is proper and convex.Further, F is lower semicontinuous with respect to the norm topology in L 2 (Ω T , R n ).Indeed, assume that the sequence ) and the dominated convergence theorem, we conclude that This concludes the proof of the lemma.

Steklov averages of the integrand
For the final approximation argument in the proof of Theorem 1.2 we need to regularize the integrand f with respect to time.To this end, extend In order to investigate convergence of the Steklov averages as ε ↓ 0, first note that specializing the proof of [11,Corollary 2.4] gives us the following result.
Lemma 2.5.Let L > 0 and assume that f : We also need the following variant of the dominated convergence theorem that can be found for example in [12,Theorem 1.20].

Mollification in time
In general, variational solutions are not admissible as comparison maps in the variational inequality (1.3), since they do not necessarily admit a derivative with respect to time.Therefore, we use the following mollification procedure with respect to time.More precisely, consider a separable Banach space X, Later on, we will mainly use A vital feature of this mollification procedure is that [v] h solves the ordinary differential equation Further, for any t o ∈ (0, T ] there holds the bound where the bracket [. ..] 1 r has to be interpreted as For maps v ∈ L r (0, T ; X) with ∂ t v ∈ L r (0, T ; X) we have the following assertion.
Lemma 2.9.Let X be a separable Banach space and r ≥ 1. Assume that v ∈ L r (0, T ; X) with ∂ t v ∈ L r (0, T ; X).Then, for the mollification in time defined by The bounded slope condition for parabolic equations Page 11 of 34 76 the time derivative can be computed by

Properties of variational solutions
As mentioned in the introduction, besides variational solutions in the sense of Definition 1.1, we consider variational solutions of the so-called gradient constrained obstacle problem to (1.1).They enjoy the same basic properties as variational solutions to the unconstrained Cauchy-Dirichlet problem to (1.1) and proofs will be given in a unified way in this section.
Let L ∈ (0, ∞].We define the following class of functions that are L-Lipschitz in space ) is a variational solution in Ω τ for any τ > 0, u is called a global variational solution or variational solution in Ω ∞ to the gradient constrained Cauchy-Dirichlet problem associated with (1.1) and u o .

Continuity with respect to time
In Definitions 1.1 and 3.1 we require that variational solutions are contained in the space C 0 ([0, T ]; L 2 (Ω)).However, this is already implied if u satisfies a variational inequality for a.e.τ ∈ [0, T ].More precisely, we have the following Lemma, which will be applied with L = ∞ in Sect.6.

Lemma 3.2. Let Ω ⊂ R n be open and bounded and T ∈ (0, ∞) and assume that
respectively.Suppose that u satisfies the variational inequality L. Schätzler and J. Siltakoski for almost all τ ∈ (0, T ) Proof.The proof is similar to that of Lemma 2.6 in [28] except for the estimate of the second integral in (3.3) below.Denote by [u] h the time mollification of u with initial values u o as defined in (2.7).In particular, observe that 2), taking the essential supremum over τ ∈ (0, T ) and recalling that (

Localization in time
Here, we show that a variational solution in a space-time cylinder Ω T is also a solution in any sub-cylinder Ω τ , τ ∈ (0, T ).
where ξ θ v has been extended to Ω T by zero and [u] h is defined according to (2.7) with initial datum u o .Then we have and therefore we may use v θ as a comparison map for u in the variational inequality.This yields 4) The first term on the right-hand side of (3.4) is identical to the one in [7,Equation (3.2)] and can be estimated in the same way to obtain lim sup The second term on the right-hand side of (3.4) is given by Since we know that by (2.1) we find that Combining the preceding estimates we arrive at Further, we have that D[u] h → Du pointwise almost everywhere in Ω T as h ↓ 0 (up to a subsequence) and that L. Schätzler and J. Siltakoski NoDEA Therefore, assumption (2.1), the fact that Ω is bounded and the dominated convergence theorem imply that lim h↓0 Ω×(τ,T ) Hence, using that ∂ t [u] h ([u] h − u) ≤ 0 and letting h ↓ 0 in (3.5), we obtain the desired inequality

The initial condition
As a consequence of the localization in time principle, we find that variational solutions attain the initial datum u o in the C 0 -L 2 -sense.The precise statement is as follows.

Comparison principle
The following comparison principle ensures in particular that variational solutions to the problems considered in the present paper are unique.
and suppose that u and ũ are variational solutions to (1.1)

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The bounded slope condition for parabolic equations Page 15 of 34 76 Proof.Let τ ∈ (0, T ].By Lemma 3.3, the functions u and ũ are variational solutions in K L (Ω τ ).Consider the functions where [u] h and [ũ] h denote the mollifications of u and ũ according to (2.7) with initial values u(0) ∈ W 1,∞ (Ω) and ũ(0) ∈ W 1,∞ (Ω), respectively.Since the boundary values attained by u and ũ are independent of time, we have ). Therefore we may use v and w as comparison functions in the variational inequalities of u and ũ, respectively.Adding the resulting inequalities and using that [u] Using the identities Therefore, taking into account that Furthermore, using [u] h as a comparison function for u and omitting the boundary term at time τ on the right-hand side of the variational inequality, we obtain and a similar inequality holds for ũ.Observe also that Combining the estimates (3.7), (3.8) and (3.9) with (3.6) we arrive at By the same argument as in the end of the proof of Lemma 3.3 involving the dominated convergence theorem, the integral on the right-hand side of (3.10) vanishes in the limit Hence, taking the limit h ↓ 0 in (3.10), we infer which implies that u ≤ ũ in Ω τ .Since τ was arbitrary, the claim follows.

Maximum principle and localization in space for regular solutions
In this section, we consider more regular variational solutions u satisfying ∂ t u ∈ L 2 (Ω T ).As a consequence, u is directly admissible as comparison map in its variational inequality without regularization with respect to the time variable.Further, due to the requirements of the proof of the existence result in Sect.5, we will take time-dependent boundary values u| Ω×(0,T ) into account here.In particular, the proof of the comparison principle in Theorem 3.5 is easily adapted to allow time-dependent boundary values if ∂ t u and ∂ t ũ are contained in L 2 (Ω T ) by using min(u, ũ) and max(u, ũ) as comparison maps in the variational inequalities satisfied by u and ũ, respectively, and proceeding in a similar way as above.However, most arguments can be simplified, since mollification with respect to time is not necessary in the present situation.This allows us to deduce the following maximum principle.
Finally, assume that for any τ ∈ (0, T ] the function u satisfies the variational inequality , where we used that (v − u)(0) = (w − û)(0) = 0 and that the terms with f cancel one another.As τ was arbitrary, we obtain Since the reverse inequality holds by continuity, this proves the claim.
Then for any domain Ω ⊂ Ω and any τ ∈ (0, T ], the variational inequality ) and v = u on ∂Ω × (0, τ).Proof.By Lemma 3.3 the function u| Ωτ is a variational solution to (1.1) in the function space K L uo (Ω τ ).Observe that is an admissible comparison function for u| Ωτ in the variational inequality.
Inserting w into the variational inequality with T replaced by τ immediately yields (3.13).

Existence for the gradient constrained problem for regular integrands
In this section, we are concerned with integrands that admit a time derivative.More precisely, we consider f : The aim of this section is to prove the following existence result.

Consider a boundary datum
Further, assume that the integrand f : [0, T ] × R n → R satisfies hypothesis (4.1).Then, there exists a variational solution u ∈ K L uo (Ω T ) to the gradient constrained problem in the sense of Definition 3.1.Further, there holds ∂ t u ∈ L 2 (Ω T ) with the quantitative bound We prove Theorem 4.1 via the method of minimizing movements.The proof is divided into five steps.

A sequence of minimizers to elliptic variational functionals
Fix a step size h := T m for some m ∈ N and consider times ih, i = 0, . . ., m.
Further, for i = 1, . . ., m, u i is defined as the minimizer of the elliptic variational functional The existence of a minimizer to F i in this class is ensured by the direct method in the calculus of variations.More precisely, note that A = ∅, since u o ∈ A, and consider a minimizing sequence to F i in A, i.e. a sequence (u i,j Further, by definition of A and Rellich's theorem there exists a limit map u i ∈ A and a (not relabelled) subsequence such that Since the functional is proper, convex and lower semicontinuous with respect to strong convergence in W 1,2 (Ω), it is also lower semicontinuous with respect to weak convergence in W 1,2 (Ω), see [11,Corollary 2.2].Therefore, we obtain that

Energy estimates
Since u i−1 ∈ A is an admissible comparison map for the minimizer u i and f fulfills (4.1) 3 , we have that L. Schätzler J. Siltakoski

NoDEA
Summing up the preceding inequalities from i = 1 to i = m, we find that Subtracting the first term on the left-hand side, we conclude that

The limit map
In the following we denote the step size by h m in order to emphasize the dependence on m.First, we join the minimizers u i to a map that is piecewise constant with respect to time.More precisely, we define Observe that the sequence Therefore, there exists a subsequence K ⊂ N and a limit map In order to prove that u has a time derivative, we consider the linear interpolation of minimizers ũ(m) : Ω × (−h m , T ] → R given by ũ(m) (t) := u o for t ∈ (−h m , 0] and Similar arguments as above ensure that ũ(m) m∈N is bounded in L ∞ (Ω T ) and that Dũ (m)  L ∞ (ΩT ,R n ) ≤ L for any m ∈ N.Moreover, by the energy bound (4.2) we obtain that

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The bounded slope condition for parabolic Page 21 of 34 76 Hence, ũ(m) m∈N is bounded in W 1,2 (Ω T ).By Rellich's theorem we conclude that there exists a subsequence still labelled K and a limit map ũ Together with (4.5) Finally, by lower semicontinuity with respect to weak convergence, (4.4) gives us the claimed bound

Minimizing property of the approximations
First, define piecewise constant approximations of the integrand by We claim that u (m) is a minimizer of the functional in the class of functions

Indeed, consider an arbitrary map
by the minimizing property of u i with respect to F i in the class A we find that A straightforward computation shows that this is equivalent to with s ∈ (0, 1) as comparison map and using the convexity of ξ → f (t, ξ) for all t ∈ [0, T ], we obtain that Reabsorbing the first term on the right-hand side into the left-hand side, dividing the resulting inequality by s and taking the limit s ↓ 0 gives us that Next, assume without loss of generality that v(0 Inserting this into the preceding inequality and recalling that v(t) = v(0) for t ∈ (−h m , 0], we infer

Variational inequality for the limit map
We fix an arbitrary map v ∈ K L uo (Ω T ) with ∂ t v ∈ L 2 (Ω T ).Thus, in particular we have that v ∈ A T , so v is an admissible comparison map in (4.6).Our goal is to pass to the limit K m → ∞ in (4.6) in order to deduce the variational inequality (3.1) for u.To this end, we consider the terms separately.First, we write the first term on the left-hand side of (4.6) as Therefore, this term vanishes in the limit m → ∞.Joining the preceding estimates, we conclude that Repeating the estimates in the penultimate inequality with u (m) replaced by v, for the first term on the right-hand side of (4.6) we find that Finally, by the fact that v ∈ C 0 ([0, T ]; L 2 (Ω)) and by (4.3) 2 , we obtain that was arbitrary, we have shown that u ∈ K L uo (Ω T ) is the desired variational solution.

Existence for the unconstrained problem for regular integrands
In this section we show the existence of variational solutions to the unconstrained problem under the regularity condition (4.1) provided that the initial and boundary datum satisfies the bounded slope condition.To this end, we need the following lemma, whose proof is similar to that of [7,Lemma 7 Combining this with the preceding estimate, we obtain (5.1).It remains to show that u is a variational solution to the unconstrained problem.Let w ∈ K ∞ uo (Ω T ) with ∂ t w ∈ L 2 (Ω T ) and choose the comparison map v := u + s(w − u) for 0 < s 1; in particular, since Q < L, for s small enough we have that Thus v is an admissible comparison function for the gradient constrained problem and we obtain that Reabsorbing the integral with f (t, Du) to the left-hand side and dividing by s, we see that u satisfies the variational inequality (1.3).Thus u is a variational solution in the sense of Definition 1.1.

Existence for the unconstrained problem for general integrands
In this section we finish the proof of Theorem 1.2.Note that we only need to consider the case T < ∞.Indeed, assume that for any τ ∈ (0, ∞) we have constructed a variational solution with initial and boundary datum u o in the Further, for any ε > 0 the derivative of f ε with respect to the time variable is given by Combining this with (2.1), for any L > 0 we have that . Hence, for any ε > 0, the integrand f ε fulfills assumption (4.1).By Theorem 5.2 we conclude that for any ε > 0 there exists a variational solution u ε ∈ K ∞ uo (Ω T ) to the Cauchy-Dirichlet problem associated with f ε in the sense of Definition 1.1 satisfying the bound Together with the fact that u ε = u o on ∂Ω × (0, T ), this implies in particular that the sequence (u ε ) ε>0 is bounded in L ∞ (Ω T ).Thus, there exists a (not relabelled) subsequence and a limit map u ∈ L ∞ (Ω T ) such that u = u o on ∂Ω × (0, T ), It remains to show that u is a variational solution to the Cauchy-Dirichlet problem associated with f in the sense of Definition 1.1.To this end, note that u ε satisfies the variational inequality for any τ ∈ [0, T ]∩R and any comparison map v ∈ K ∞ uo (Ω τ ) with ∂ t v ∈ L 2 (Ω τ ).In the following, we pass to the limit ε ↓ 0 in (6.2).In order to treat the lefthand side, we rewrite

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The bounded slope condition for parabolic equations For the second term on the right-hand side of (6.2), by Lemma 2.7 we conclude that as ε ↓ 0. Finally, (6.1) 2 shows that ) is a variational solution associated with the integrand f in the sense of Definition 1.1.Finally, by the comparison principle in Theorem 3.5, u is unique.This concludes the proof of Theorem 1.2.

Continuity in time (Proof of Theorem 1.3)
To prove Theorem 1.Since by (1.4) we have that it follows from (2.1) that the sequence of mappings (x, t) → f (t, Dv h (x, t)) has an integrable dominant independent of h.Therefore by the dominated convergence theorem, we conclude that Further, by integration by parts and the convergence assertions from Lemmas 2.8 and 2.9, we find that  To prove (7.3), we first note that since Ω is a convex domain, there exist positive constants R(Ω) and C(Ω) such that for any r ∈ (0, R) and x 0 ∈ Ω, the set Ω ∩ B r (x 0 ) contains a ball of radius r/C(Ω).Then we assume that Q r with r < R is given and denote B r := B r (x 0 ), t 1 := max(t 0 − r 2 , 0), t 2 := min(t 0 + r 2 , T ) so that Q r = (B r ∩ Ω) × (t 1 , t 2 ).We fix a non-negative weight function η ∈ C ∞ 0 (B r ∩ Ω) such that − Br∩Ω η dx = 1 and η L ∞ (Ω) + r Dη L ∞ (Ω;R n ) ≤ c(n, Ω).
For the second assertion, we have used that B r ∩ Ω contains a ball of size r/C(Ω).Since B r ∩ Ω is convex, the Poincaré inequality holds for any v ∈ W 1,2 (B r ∩ Ω), see for example [1].An application of Hölder's and Minkowski's inequalities on the above further yields To estimate I 3 , we apply (7.5) to obtain that The same estimate holds for I 2 since by Hölder's inequality we have that denote a boundary datum such that the bounded slope condition with some positive constant Q (see Definition 2.1 below) is fulfilled for U o := u o | ∂Ω .Then, there exists a unique variational solution u to the Cauchy-Dirichlet problem associated with (1.1) and u o in Ω T .Moreover, u satisfies the gradient bound Du)| dxdt.(3.3)Furthermore, we have that D[u] h → Du almost everywhere in Ω T as h ↓ 0 (up to a subsequence) and that |D[u] h | ≤ |Du o | + sup ΩT |Du|.Therefore, by (2.1) and the dominated convergence theorem we find that the second integral in (3.3) vanishes in the limit h ↓ 0. Hence, we have shown that [u] h → u in L ∞ (0, T ; L 2 (Ω)).Combining this with the fact that [u] h ∈ C 0 ([0, T ]; L 2 (Ω)), it follows that also u ∈ C 0 ([0, T ]; L 2 (Ω)).

Proof.
By Lemma 3.3, the function u is a variational solution in any smaller cylinder Ω τ , τ ∈ (0, T ].Using v : Ω τ → R, v(x, t) := u o (x) as a comparison function for u and taking (2.1) with M

7 .
(Localization in space) Let T ∈ (0, ∞), assume that Ω ⊂ R n is open and bounded, and that
3, we begin by verifying that the unique variational solution u to the Cauchy-Dirichlet problem associated with (1.1) and u o in Ω T is a weak solution to (1.1) in Ω T .To this end, let ϕ ∈ C ∞ 0 (Ω T ) be a test function.We want to show that ΩT u∂ t ϕ dxdt = ΩT D ξ f (t, Du) • Dϕ dxdt.(7.1) L. Schätzler and J. Siltakoski NoDEA We set v h := [u] h + s[ϕ] h , where in the convolution we use the starting values u o and ϕ(0) = 0 for u and ϕ, respectively.Using v h as a comparison function in (1.3) and omitting the boundary term at T , we obtain that 0 ≤ ΩT ∂ t v h (v h − u) dxdt + ΩT f (t, Dv h ) − f (t, Du) dxdt.(7.2) lim h↓0 ΩT f (t, Dv h ) dxdt = ΩT f (t, Du + sDϕ) dxdt.

Definition 2.1. A function U : ∂Ω → R satisfies the bounded slope condition with constant Q > 0 if for any x o ∈ ∂Ω there exist two affine functions w ± xo : R n → R with Lipschitz constants [w ± xo
it is crucial that there exist affine comparison functions below and above the initial/boundary datum u o coinciding with u o in a point x o ∈ ∂Ω.This is ensured by applying the following bounded slope condition to u o | ∂Ω .

Lemma 2.2. Let
7, Lemma 2.3].It states that if u o is Lipschitz and u o | ∂Ω satisfies the bounded slope condition, then u o can be L. Schätzler and J. Siltakoski NoDEA squeezed between two affine functions that touch u o at a given boundary boundary point and the Lipschitz constant of these affine functions is bounded by either the Lipschitz constant of u o or the constant in the bounded slope condition.u o ∈ C 0,1 (Ω) with Lipschitz constant [u o ] 0,1 ≤ Q 1 such that the restriction U := u o | ∂Ω satisfies the bounded slope condition with constant Q 2 .Then for any x o ∈ ∂Ω there exist two affine functions w ± xo with [w ± xo is called a variational solution to the gradient constrained Cauchy-Dirichlet problem associated with (1.1) and u o in Ω T if and only if the variational inequality .10) .1].It states that affine functions independent of time are variational solutions to (1.1) with respect to their own initial and lateral boundary values.Let Ω be open and bounded.Assume thatf : [0, T ] × R n → R satisfies (1.2).Let w(x, t) := a + ξ • x with constants a ∈ R and ξ ∈ R n be an affine function independent of time.Then w is a variational solution in the sense of Definition 1.1 in K ∞ w (Ω T ).With the preceding lemma at hand, we are able to prove the following.Let T ∈ (0, ∞), assume that Ω ⊂ R n is open, bounded and convex, and that the integrand f :[0, T ] × R n → R satisfies (4.1).Consider u o ∈ W 1,∞ (Ω) such that Du o L ∞ (Ω,R n ) ≤ Qand suppose that u o | ∂Ω satisfies the bounded slope condition with the same parameter Q.Then there exists a variational solution u ∈ K ∞ uo (Ω T ) to (1.1) in the sense of Definition 1.1.Further, we have the quantitative bound .1) Proof.Let L > Q.By Theorem 4.1 there exists a variational solution u ∈ K L uo (Ω T ) with ∂ t u ∈ L 2 (Ω T ) to the gradient constrained problem in the sense of Definition 3.1.We begin by proving the Lipschitz bound (5.1) and then show that u is in fact already a solution to the unconstrained problem.Fix x o ∈ ∂Ω and denote by w ± xo the affine functions from Lemma 2.2.In particular we have w − xo ≤ u o ≤ w + xo .Since by Lemma 5.1 the functions w− − u o (x o )| ≤ Q |x − x o | for all (x, t) ∈ Ω T .Since x o ∈ ∂Ω was arbitrary, we obtain that |u(x, t) − u o (x o )| |x − x o | ≤ Q for all x o ∈ ∂Ω, (x, t) ∈ Ω T .(5.2) Consider x 1 , x 2 ∈ Ω, x 1 = x 2 ,t ∈ (0, T ) and set y := x 2 − x 1 .Define the shifted set Ω T := (x − y, t) ∈ R n+1 : (x, t) ∈ Ω T and the shifted function u y : Ω T → R by Then u y is a variational solution in K L ( Ω T ).Since ∂ t u, ∂ t u y ∈ L 2 ((Ω ∩ Ω) T ) by the spatial localization principle in Lemma 3.7, the functions u and u y both satisfy variational inequality (3.11) from Lemma 3.6 in (Ω ∩ Ω) T .Therefore by Lemma 3.6 there exists (x o , t o ) ∈ ∂ P ((Ω ∩ Ω)) T such that |u(x 1 , t) − u y (x 1 , t)| ≤ |u(x o , t o ) − u y (x o , t o )| .By definition of y and u y , this yields |u(x 1 , t) − u(x 2 , t)| ≤ |u(x o , t o ) − u(x o + y, t o )| .Since either t o = 0 or one of the points x o or x o + y belongs to ∂Ω, it follows from the assumption Du Definition 1.1 such that the gradient bound(1.4)holds in Ω τ .Let 0 < τ 1 < τ 2 < ∞ and denote by u 1 and u 2 the variational solutions in Ω τ1 and Ω τ2 , respectively.By the localization principle with respect to time in Lemma 3.3, u 2 is also a variational solution in Ω τ1 .Further, u 1 and u 2 coincide in Ω τ1 by the comparison principle in Theorem 3.5.Therefore, a unique global variational solution in the sense of Definition 3.1 can be constructed by taking an increasing sequence of times (τ i ) i∈N with lim i→∞ τ i = ∞.Thus we suppose that T < ∞.For ε > 0 we define the Steklov average f L. Schätzler and J. Siltakoski NoDEA sense of Page 27 of 34 76 By (6.1) 3 and Lemma 2.4 we obtain that Du ε ) dxdt.Further, for M := max{Q, Du o L ∞ (Ω,R n ) } we find that Ωτf ε (t, Du ε ) − f (t, Du ε ) dxdt ≤ |Ω|