The bounded slope condition for parabolic equations with time-dependent integrands
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ The bounded slope condition for parabolic equations with time-dependent integrands © 2023 The Author(s) Published version Schätzler, Leah; Siltakoski, Jarkko Schätzler, L., & Siltakoski, J. (2023). The bounded slope condition for parabolic equations with time-dependent integrands. Nodea: Nonlinear Differential Equations and Applications, 30, Article 76. https://doi.org/10.1007/s00030-023-00876-6 2023
Nonlinear Differ. Equ. Appl. (2023) 30:76 c 2023 The Author(s) https://doi.org/10.1007/s00030-023-00876-6 Nonlinear Differential Equations and Applications NoDEA The bounded slope condition for parabolic equations with time-dependent integrands Leah Sch¨atzler and Jarkko Siltakoski Abstract. In this paper, we study the Cauchy–Dirichlet problem ∂tu−div (Dξf(t, Du)) = 0 in ΩT, u=uoon ∂PΩT, where Ω ⊂Rnis a convex and bounded domain, f:[0,T]×Rn→Ris L1-integrable in time and convex in the second variable. Assuming that the initial and boundary datum uo: Ω →Rsatisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable. Mathematics Subject Classification. 35A01, 35K61, 35K86, 49J40. Keywords. Existence, Parabolic equations, Bounded slope condition, Lipschitz solutions, Time-dependent integrand. 1. Introduction and results It follows from classical theory [15,17,18,26,29](seealso[14, Chapter 1]) that any variational functional F:W1,∞(Ω) →Rof the form F(v):=Ω f(Dv)dx, where f:Rn→Ris convex and Ω ⊂Rnis a convex domain, admits a unique Lipschitz continuous minimizer in the class {v∈W1,∞(Ω) : v=voon ∂Ω} provided that the boundary datum vosatisfies the bounded slope condition (see Definition 2.1). Modern elliptic results involving one-sided bounded slope conditions or more general integrands include for example [2–4,10,13,22–24]. 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 B¨ogelein, Duzaar, Marcellini and Signoriello [7]. They considered the Cauchy–Dirichlet problem 0123456789().: V,-vol
76 Page 2 of 34 L. Schätzler and J. Siltakoski NoDEA ∂tu−div (Df(Du)) = 0 in ΩT, u=uoon ∂PΩT, where ΩT:= Ω ×(0,T) with Ω ⊂Rnand T∈(0,∞] denotes a space-time cylinder and ∂PΩT:= ∂Ω×(0,T)∪(Ω ×{0}) its parabolic boundary. Given a Lipschitz continuous initial and boundary datum uothat 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 ffulfills an additional p-coercivity condition with some p>1, B¨ogelein and Stanin [8] obtained the local Lipschitz continuity of variational solutions in space and time under the assumption that uois convex and Lipschitz continuous. Further, global continuity of uwas 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 H¨older continuous even if the convexity assumption on uois 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 ∂tu−div (Df(Du)) + Dug(x, u)=0 in Ω T, u=uoon ∂PΩT, where fis convex and p-coercive with some p>1 and the lower-order term gsatisfies 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∈C2is 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 fwith 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 ∂tu−div(Dξf(t, Du)) = 0 in ΩT,(1.1) where Ω ⊂Rnis a convex and bounded domain and T∈(0,∞]. The integrand f:[0,T]×Rn→Ris assumed to be a Carath´eodory function that satisfies the following assumptions: ξ→ f(t, ξ)isconvexinRnfor a.e. t∈[0,T], t→ f(t, ξ)∈L1(0,τ) for all ξ∈Rnand τ∈(0,T]∩R.(1.2) In particular, for any L>0andτ∈(0,T]∩Rthe map t→ max|ξ|≤L|f(t, ξ)| belongs to L1(0,τ) (see Sect. 2.3 below). Therefore, for any τ∈(0,T]∩Rand V∈L∞(ΩT,Rn)wehavethat
NoDEA The bounded slope condition for parabolic equations Page 3 of 34 76 ΩT |f(t, V )|dxdt<∞. We emphasize that t→ f(t, ξ) is neither assumed to be continuous nor weakly differentiable. Examples of admissible integrands are functionals with linear growth such as the area integrand f(ξ)=1+|ξ|2, integrands with exponential growth like f(ξ) = exp(|ξ|2), Orlicz type functionals such as f(ξ)=|ξ|log(1+|ξ|)and time-dependent combinations thereof like f(t, ξ)=χ[0,to]f1(ξ)+χ(to,T ]f2(ξ) or more general f(t, ξ)=m i=1 ai(t)fi(ξ) for functions ai∈L1(0,T), i= 1,...,m. 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 K∞:= {v∈L∞(ΩT)∩C0([0,T]; L2(Ω)) : Dv ∈L∞(ΩT,Rn)}. Further, we denote the subclass related to time-independent boundary values uo∈W1,∞(Ω) by K∞ uo:= {v∈K∞(ΩT):v=uoon the lateral boundary ∂Ω×(0,T)}. Definition 1.1. (Variational solutions) Assume that f:[0,T]×Rn→Rsatisfies (1.2) and consider a boundary datum uo∈W1,∞(Ω). In the case T∈ (0,∞) a map u∈K∞ uo(ΩT) is called a variational solution to the Cauchy– Dirichlet problem associated with (1.1)anduoin ΩTif and only if the variational inequality ΩT f(t, Du)dxdt≤ΩT ∂tv(v−u)+f(t, Dv)dxdt +1 2v(0) −uo2 L2(Ω) −1 2(v−u)(T)2 L2(Ω) (1.3) holds true for any comparison map v∈K∞ uo(ΩT) with ∂tv∈L2(ΩT). If T=∞ and u∈K∞ uo(Ω∞) is a variational solution in Ωτfor any τ∈(0,∞), uis called aglobal variational solution or variational solution in Ω∞to the Cauchy– Dirichlet problem associated with (1.1)anduo. 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⊂Rnbe an open, bounded and convex set and T∈(0,∞]. Assume that f:[0,T]×Rn→Rsatisfies hypotheses (1.2). Further, let uo∈ W1,∞(ΩT)denote a boundary datum such that the bounded slope condition with some positive constant Q(see Definition 2.1 below) is fulfilled for Uo:= uo|∂Ω. Then, there exists a unique variational solution uto the Cauchy–Dirichlet problem associated with (1.1)and uoin ΩT. Moreover, usatisfies the gradient bound DuL∞(ΩT,Rn)≤max{Q, DuoL∞(Ω,Rn)}.(1.4)
76 Page 4 of 34 L. Schätzler and J. Siltakoski NoDEA Furthermore, we show that variational solutions to (1.1) are weak solutions and consequently, they are 1/2-H¨older continuous in time provided that the map ξ→ f(t, ξ)isC1and uniformly locally Lipschitz in the following sense: For each L>0, there exists a constant ML>0 such that sup t∈(0,T ) |Dξf(t, ξ)|<M Lfor all ξ∈BL(0).(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 C1(Rn)foralmostallt∈(0,T)and satisfies (1.5). Then the unique variational solution uto the Cauchy–Dirichlet problem associated with (1.1)and uois a weak solution (see (7.1)). Further, it is contained in the space of H¨older continuous functions C0;1,1/2(ΩT). 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 uto 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 fensures that uhas a weak time derivative in L2(ΩT). Next, under the same regularity assumptions on fas in the first step, a standard argument exploiting the bounded slope condition and the maximum principle yields the uniform gradient bound (1.4)foru. Choosing Llarge enough, this in turn allows us to deduce that uis 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 fε. Since fεadmits a weak time derivative, by the results mentioned in the preceding paragraph there exists a solution uεto the Cauchy–Dirichlet problem associated with fεin the sense of Definition 1.1. Moreover, since for each ε>0the solution uεsatisfies the gradient bound (1.4)anduε=uoon ∂Ω×(0,T), there exists a limit map u∈L∞(ΩT) such that uε→uuniformly and Duε∗ 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 C1assumption 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 Poincar´e inequality for variational solutions. The H¨older continuity then follows from the Campanato space characterization of H¨older continuity by Da Prato [9]. The paper is organized as follows. Section 2contains preliminary definitions and basic observations about the integrand. In Sect. 3we prove certain properties of variational solutions that are required in later sections, including the comparison and maximum principles. Under additional regularity assumptions on fwe use the method of minimizing movements to prove the existence
NoDEA The bounded slope condition for parabolic equations Page 5 of 34 76 of variational solutions to the gradient constrained problem in Sect. 4and in Sect. 5we consider the unconstrained problem. Finally, in Sect. 6we consider general integrands and finish the proof of Theorem 1.2 and H¨older continuity in time is proven in Sect. 7under additional regularity assumptions. 2. Preliminaries 2.1. Notation Throughout the paper, for p∈[1,∞]andm∈Nthe space Lp(Ω,Rm) denotes the usual Lebesgue space (we omit Rmif m=1)andW1,p(Ω) and W1,p 0(Ω) denote the usual Sobolev spaces. If Ω is a bounded Lipschitz domain, W1,∞(Ω) can be identified with the space C0,1(Ω) of functions v:Ω →Rthat are Lipschitz continuous (with Lipschitz constant [v]0,1=DvL∞(Ω,Rn))upto 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 Xand an integrability exponent p∈[1,∞]we write Lp(0,T;X) for the space of Bochner measurable functions v:[0,T]→X with t→v(t)X∈Lp(0,T). Moreover, C0([0,T]; X) is defined as the space of the continuous functions v:[0,T]→X. For maps vdefined in ΩTwe also use the short notation v(t) for the partial map x→ v(x, t) defined in Ω. Finally, for a set A⊂Rm, the characteristic function χA:Rm→{0,1}is given by χA(x)=1ifx∈Aand χA(x)=0else. 2.2. Bounded slope condition In the proof of the existence result in Sect. 5it is crucial that there exist affine comparison functions below and above the initial/boundary datum uo coinciding with uoin a point xo∈∂Ω. This is ensured by applying the following bounded slope condition to uo|∂Ω. Definition 2.1. A function U:∂Ω→Rsatisfies the bounded slope condition with constant Q>0 if for any xo∈∂Ω there exist two affine functions w± xo:Rn→Rwith Lipschitz constants [w± xo]0,1≤Qsuch that w− xo(x)≤U(x)≤w+ xo(x) for any x∈∂Ω, w− xo(xo)=U(xo)=w+ xo(xo). Note that unless Uitself 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 C2-domain Ω and v∈C2(Rn) the restriction U=v|∂Ωfulfills 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 uothat 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 [7, Lemma 2.3]. It states that if uois Lipschitz and uo|∂Ωsatisfies the bounded slope condition, then uocan be
76 Page 6 of 34 L. Schätzler and J. Siltakoski NoDEA squeezed between two affine functions that touch uoat a given boundary boundary point and the Lipschitz constant of these affine functions is bounded by either the Lipschitz constant of uoor the constant in the bounded slope condition. Lemma 2.2. Let uo∈C0,1(Ω) with Lipschitz constant [uo]0,1≤Q1such that the restriction U:= uo|∂Ωsatisfies the bounded slope condition with constant Q2. Then for any xo∈∂Ωthere exist two affine functions w± xowith [w± xo]0,1≤ max{Q1,Q 2}such that w− xo(x)≤uo(x)≤w+ xo(x)for any x∈Ω, w− xo(xo)=uo(xo)=w+ xo(xo). 2.3. Dominating functions for the integrand Observe that for any L>0 the map t→ max|ξ|≤Lf(t, ξ) is measurable, since we have that max|ξ|≤Lf(t, ξ) = maxξ∈BL(0)∩Qnf(t, ξ) and the maximum of countably many measurable functions is measurable. The same holds true for t→ min|ξ|≤Lf(t, ξ). In the following lemma, we show that they are contained in L1(0,T). Lemma 2.3. Let T∈(0,∞)and assume that f:[0,T]×Rn→Rsatisfies (1.2). Then, for any L>0there exists a function gL∈L1(0,T)such that |f(t, ξ)|≤gL(t)for all t∈(0,T)and ξ∈BL(0).(2.1) Proof. First, we show that for any L>0, we have that t→ max |ξ|≤Lf(t, ξ)∈L1(0,T).(2.2) To this end, fix ξ1,...,ξ n+1 ∈Rnsuch that the closed ball BL(0) is a subset of the simplex Δ:=ξ∈Rn:ξ= n+1 i=1 λiξiwith 0 ≤λi≤1,i=1,...,n+1, n+1 i=1 λi=1 . Note that for any t∈[0,T] such that Rnξ→ f(t, ξ) is convex, the mapping ξ→ f(t, ξ) attains its maximum in one of the points ξ1,...,ξ n+1. Hence, for a.e. twe obtain that f(t, 0) ≤max |ξ|≤Lf(t, ξ)≤ n+1 i=1 |f(t, ξi)|. Since the maps t→ f(t, 0) and t→ f(t, ξi), i=1,...,n+ 1, belong to L1(0,T)by(1.2)2, this implies (2.2). Next, we fix L>0 and prove t→ min |ξ|≤Lf(t, ξ)∈L1(0,T).(2.3) Consider t∈[0,T] such that ξ→ f(t, ξ) is convex. Then, there exist ξmin,ξ max ∈BL(0) such that f(t, ξmin)=min|ξ|≤Lf(t, ξ)andf(t, ξmax)=max|ξ|≤Lf(t, ξ). Assume that ξmin =ξmax (otherwise, ξ→ f(t, ξ) is constant in BL(0) and
NoDEA The bounded slope condition for parabolic equations Page 7 of 34 76 thus f(t, 0) = min|ξ|≤L)f(t, ξ)). First, note that for C:= 1 2L(f(t, ξmax)− f(t, ξmin)) ∈(0,∞), we find that f(t, ξmin)≤f(t, ξmax)−C|ξmax −ξmin|. Furthermore, since ξ→ f(t, ξ)isconvexinRn, its subdifferential at ξmax is non-empty [11, Proposition 5.2], i.e. there exists η=η(ξmax)∈Rnsuch that f(t, ξ)≥f(t, ξmax)+η·(ξ−ξmax) for any ξ∈Rn. In particular, we have that f(t, ξmin)≥f(t, ξmax)+η·(ξmin −ξmax) =f(t, ξmax)+cos(α)|η||ξmin −ξmax|, where αdenotes the angle between ηand ξmin −ξmax. Together, the preceding two inequalities imply that cos(α)|η|≤−C. Next, choose s>1 such that ξo:= ξmin +s(ξmax −ξmin)∈∂BL+1(0). Note that the vector ξo−ξmax =(1−s)(ξmin −ξmax) points in the opposite direction as ξmin −ξmax. Therefore, the angle between ηand ξo−ξmax is π−α. Using the facts that cos(π−α)=−cos(α)and|ξo−ξmax|≥1, the preceding inequality and the definition of C, we conclude that max |ξ|≤L+1 f(t, ξ)≥f(t, ξo)≥f(t, ξmax)+η·(ξo−ξmax) =f(t, ξmax)−cos(α)|η||ξo−ξmax| ≥f(t, ξmax)+C = max |ξ|≤Lf(t, ξ)+ 1 2L(max |ξ|≤Lf(t, ξ)) −min |ξ|≤Lf(t, ξ))). This is equivalent to (2L+ 1) max |ξ|≤Lf(t, ξ)−2Lmax |ξ|≤L+1 f(t, ξ)≤min |ξ|≤Lf(t, ξ)≤max |ξ|≤Lf(t, ξ), which holds for almost every t∈[0,T]. Since we have already shown that t→ max|ξ|≤Lf(t, ξ)andt→ max|ξ|≤L+1 f(t, ξ) are contained in L1(0,T), the preceding inequality proves (2.3). The claim of Lemma 2.3 follows by combining (2.2) and (2.3). 2.4. Lower semicontinuity In the course of the paper we will need the following result on the lower semicontinuity of integrals involving fwith respect to the weak∗topology of L∞(ΩT,Rn). Lemma 2.4. Let Ω⊂Rnbe a bounded open set and 0<T <∞.Assume that f:[0,T]×Rn→Rsatisfies (1.2). Then, for any sequence (Vi)i∈N⊂ L∞(ΩT,Rn)and V∈L∞(ΩT,Rn)such that Vi∗ V weakly∗in L∞(ΩT,Rn) as i→∞we have that ΩT f(t, V )dxdt≤lim inf i→∞ ΩT f(t, Vi)dxdt.
76 Page 8 of 34 L. Schätzler and J. Siltakoski NoDEA Proof. Consider an arbitrary sequence (Vi)i∈N⊂L∞(ΩT,Rn) and a limit map V∈L∞(ΩT,Rn) such that Vi∗ V weakly∗in L∞(ΩT,Rn)asi→∞.First, note that (Vi)i∈Nis bounded in L∞(ΩT,Rn) and set M:=supi∈NViL∞(ΩT,Rn) ≥VL∞(ΩT,Rn). We find that C:= {W∈L2(ΩT,Rn):WL∞(ΩT,Rn)≤M} is a convex subset of L2(ΩT,Rn). Therefore, since ξ→ f(t, ξ) is convex for a.e. t∈[0,T] and since ΩTf(t, W )dxdtis finite for any W∈Cby (2.1), we obtain that the functional F:L2(ΩT,Rn)→(−∞,∞] given by F[W]:=ΩTf(t, W)dxdtif W∈C, ∞else is proper and convex. Further, Fis lower semicontinuous with respect to the norm topology in L2(ΩT,Rn). Indeed, assume that the sequence (Wi)i∈N⊂ L2(ΩT,Rn) converges strongly in L2(ΩT,Rn) to a limit map W∈L2(ΩT,Rn) as i→∞. If lim infi→∞ F[Wi]=∞, the assertion F[W]≤lim infi→∞ F[Wi] holds trivially. Otherwise, there exists a subsequence K⊂Nsuch that Wi∈C for any i∈K, lim infi→∞ F[Wi] = limKi→∞ F[Wi]andWi→Wa.e. in ΩTas Ki→∞.By(2.1) and the dominated convergence theorem, we conclude that F[W] = limKi→∞ F[Wi] = lim infi→∞ F[Wi]. Therefore, Fis also lower semicontinuous with respect to the weak topology in L2(ΩT,Rn), cf. [11, Corollary 2.2]. Since ΩTis bounded, we have that ViV weakly in L2(ΩT,Rn)asi→∞and hence ΩT f(t, V )dxdt=F[V]≤lim inf i→∞ F[Vi] = lim inf i→∞ ΩT f(t, Vi)dxdt. This concludes the proof of the lemma. 2.5. Steklov averages of the integrand For the final approximation argument in the proof of Theorem 1.2 we need to regularize the integrand fwith respect to time. To this end, extend fto [0,∞]×Rnby zero if T<∞.Forε>0 define the Steklov average fε:[0,T]× Rn→Rof the extended integrand by fε(t, ξ):=− t+ε t f(s, ξ)ds. (2.4) 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>0and assume that f:Rn→Ris a convex function with fL∞(BL+1(0)) ≤C.Then,fsatisfies the local Lipschitz continuity condition |f(ξ1)−f(ξ2)|≤2C|ξ1−ξ2|for all ξ1,ξ 2∈BL(0). We also need the following variant of the dominated convergence theorem that can be found for example in [12, Theorem 1.20].
NoDEA The bounded slope condition for parabolic equations Page 15 of 34 76 Proof. Let τ∈(0,T]. By Lemma 3.3, the functions uand ˜uare variational solutions in KL(Ωτ). Consider the functions v:= min([u]h,[˜u]h)andw:= max([u]h,[˜u]h), where [u]hand [˜u]hdenote the mollifications of uand ˜uaccordingto(2.7) with initial values u(0) ∈W1,∞(Ω) and ˜u(0) ∈W1,∞(Ω), respectively. Since the boundary values attained by uand ˜uare independent of time, we have that v∈KL u(Ωτ)andw∈KL ˜u(Ωτ) with ∂tv,∂tw∈L2(Ωτ). Therefore we may use vand was comparison functions in the variational inequalities of uand ˜u, respectively. Adding the resulting inequalities and using that [u]h(0) = u(0) ≤ ˜u(0) = [˜u]h(0), we obtain 0≤Ωτ ∂tv(v−u)+∂tw(w−˜u)dxdt +Ωτ f(t, Dv)−f(t, Du)+f(t, Dw)−f(t, D˜u)dxdt −1 2(v−u)(τ)2 L2(Ω) −1 2(w−˜u)(τ)2 L2(Ω) .(3.6) Using the identities v−u= min([u]h,[˜u]h)−[u]h−(u−[u]h)=−([u]h−[˜u]h)+−h∂t[u]h, w−˜u=([u]h−[˜u]h)+−h∂t[˜u]h, we compute that ∂tv(v−u)+∂tw(w−˜u) =∂t[u]hχ{[u]h≤[˜u]h}+∂t[˜u]hχ{[˜u]h<[u]h}−[u]h−[˜u]h+−h∂t[u]h +∂t[˜u]hχ{[u]h≤[˜u]h}+∂t[u]hχ{[˜u]h<[u]h}[u]h−[˜u]h+−h∂t[˜u]h =∂t[˜u]h[u]h−[˜u]h+−∂t[u]h([u]h−[˜u]h)+−h(∂t[u]h)2−h(∂t[˜u]h)2 ·χ{[u]h≤[˜u]h} +∂t[u]h[u]h−[˜u]h+−∂t[˜u]h[u]h−[˜u]h+−h∂t[˜u]h∂t[u]h−h∂t[u]h∂t[˜u]h ·χ{[˜u]h<[u]h} ≤∂t[u]h[u]h−[˜u]h+−∂t[˜u]h[u]h−[˜u]h+−h∂t[˜u]h∂t[u]h−h∂t[u]h∂t[˜u]h ·χ{[˜u]h<[u]h} =∂t[u]h−[˜u]h[u]h−[˜u]h+−2h∂t[u]h∂t[˜u]hχ{[˜u]h<[u]h} ≤1 2∂t[u]h−[˜u]h+)2+h∂t[u]h2+∂t[˜u]h2. Therefore, taking into account that [u]h(0) = u(0) ≤˜u(0) = [˜u]h(0), we find that Ωτ ∂tv(v−u)+∂tw(w−˜u)dxdt ≤1 2 [u]h−[˜u]h+(τ) 2 L2(Ω) +Ωτ h∂t[u]h2+∂t[˜u]h2dxdt. (3.7) Furthermore, using [u]has a comparison function for uand omitting the boundary term at time τon the right-hand side of the variational inequality, we obtain
76 Page 16 of 34 L. Schätzler and J. Siltakoski NoDEA Ωτ h∂t[u]h2dxdt=−Ωτ ∂t[u]h[u]h−udxdt ≤Ωτ ft, D[u]h−f(t, Du)dxdt(3.8) and a similar inequality holds for ˜u. Observe also that f(t, Dv)−f(t, Du)+f(t, Dw)−f(t, D˜u) =χ{[u]h≤[˜u]h}ft, D[u]h+χ{[˜u]h<[u]h}ft, D[˜u]h−f(t, Du) +χ{[u]h≤[˜u]h}ft, D[˜u]h+χ{[˜u]h<[u]h}ft, D[u]h−f(t, D˜u) =ft, D[u]h−f(t, Du)+ft, D[˜u]h−f(t, D˜u).(3.9) Combining the estimates (3.7), (3.8) and (3.9) with (3.6) we arrive at −1 2 [u]h−[˜u]h+(τ) 2 L2(Ω) +1 2(v−u)(τ)2 L2(Ω) +1 2(w−˜u)(τ)2 L2(Ω) ≤2Ωτ ft, D[u]h−f(t, Du)+ft, D[˜u]h−f(t, D˜u)dxdt. (3.10) 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 h↓0. Writing v−u=−([u]h−[˜u]h)++[u]h−uand w−˜u=([u]h−[˜u]h)++[˜u]h−˜uand using that [u]h→uand [˜u]h→˜uin L∞([0,τ],L 2(Ω)) as h↓0 since u, ˜u∈C0([0,T]; L2(Ω)), we conclude that lim h↓0−1 2 [u]h−[˜u]h+(τ) 2 L2(Ω) +1 2(v−u)(τ)2 L2(Ω) +1 2(w−˜u)(τ)2 L2(Ω) =1 2 (u−˜u)+(τ) 2 L2(Ω). Hence, taking the limit h↓0in(3.10), we infer 1 2 (u−˜u)+(τ) 2 L2(Ω) ≤0, which implies that u≤˜uin Ωτ. Since τwas arbitrary, the claim follows. 3.5. Maximum principle and localization in space for regular solutions In this section, we consider more regular variational solutions usatisfying ∂tu∈L2(ΩT). As a consequence, uis 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 ∂tuand ∂t˜uare contained in L2(ΩT) by using min(u, ˜u) and max(u, ˜u) as comparison maps in the variational inequalities satisfied by uand ˜u, 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.
NoDEA The bounded slope condition for parabolic equations Page 17 of 34 76 Lemma 3.6. (Maximum principle)LetT∈(0,∞), assume that Ω⊂Rnis open and bounded, and that f:[0,T]×Rn→Rsatisfies (1.2). Consider L∈ (0,∞]and functions u, ˜u∈KL(ΩT)such that ∂tu, ∂t˜u∈L2(ΩT).Suppose moreover that Du(0)L∞(Ω,Rn)and D˜u(0)L∞(Ω,Rn)are bounded by Lif L∈ (0,∞)and finite if L=∞. Finally, assume that for any τ∈(0,T]the function usatisfies the variational inequality Ωτ f(t, Du)dxdt≤Ωτ ∂tv(v−u)+f(t, Dv)dxdt +1 2u(0) −v(0)2 L2(Ω) −1 2u(τ)−v(τ)2 L2(Ω) (3.11) whenever v∈KL(Ωτ)with ∂tv∈L2(Ωτ)and v=uon Ω×(0,τ), and that ˜u fulfills the analogical inequality. Then sup ΩT (u−˜u)= sup ∂PΩT (u−˜u). Proof. Let τ∈(0,T]. Define ˆu:= ˜u+sup ∂PΩT (u−˜u). Then ˆusatisfies the variational inequality (3.11) with its own boundary values, and u≤ˆuon ∂PΩT.(3.12) Consider the functions v:= min(u, ˆu)andw:= max(u, ˆu). Then v,w ∈ KL(Ωτ) with ∂tv,∂tw∈L2(Ωτ)andv=u,w=ˆuon ∂Ω×(0,τ). Observe also that v−u=−(u−ˆu)+and w−ˆu=(u−ˆu)+.Usingvand was comparison functions for uand ˆuin the variational inequality (3.11), we obtain 0≤Ωτ ∂tv(v−u)+∂tw(w−ˆu)dxdt +Ωτ f(t, Dv)−f(t, Du)+f(t, Dw)−f(t, Dˆu)dxdt +1 2(v−u)(0)2 L2(Ω) +1 2(w−ˆu)(0)2 L2(Ω) −1 2(v−u)(τ)2 L2(Ω) −1 2(w−ˆu)(τ)2 L2(Ω) =Ωτ 1 2∂t((u−ˆu)+)2dxdt−(u−ˆu)+(τ)2 L2(Ω) =−1 2(u−ˆu)+(τ)2 L2(Ω) , where we used that (v−u)(0) = (w−ˆu)(0) = 0 and that the terms with f cancel one another. As τwas arbitrary, we obtain u≤ˆu=˜u+sup ∂PΩT (u−ˆu)inΩ T so that sup ΩT (u−˜u)≤sup ∂PΩT (u−˜u). Since the reverse inequality holds by continuity, this proves the claim.
76 Page 18 of 34 L. Schätzler and J. Siltakoski NoDEA Lemma 3.7. (Localization in space) Let T∈(0,∞), assume that Ω⊂Rn is open and bounded, and that f:[0,T]×Rn→Rsatisfies (1.2). Consider uo∈W1,∞(Ω) and L∈(0,∞]such that DuoL∞(Ω,Rn)≤L. Suppose that uis a variational solution to (1.1)in KL uo(ΩT),L∈(0,∞](in the sense of Definition 3.1 if L<∞, in the sense of Definition 1.1 if L=∞). Moreover, suppose that ∂tu∈L2(ΩT). Then for any domain Ω⊂Ωand any τ∈(0,T], the variational inequality Ω τ f(t, Du)dxdt≤Ω τ ∂tv(v−u)+f(t, Dv)dxdt +1 2u(0) −v(0)2 L2(Ω)−1 2u(τ)−v(τ)2 L2(Ω)(3.13) holds whenever v∈KL uo(Ω τ)with ∂tv∈L2(Ωτ)and v=uon ∂Ω×(0,τ). Proof. By Lemma 3.3 the function u|Ωτis a variational solution to (1.1)inthe function space KL uo(Ωτ). Observe that w:= vin Ω τ, uin (Ω \Ω)τ, is an admissible comparison function for u|Ωτin the variational inequality. Inserting winto the variational inequality (3.1)ifL<∞(or (1.3)ifL=∞) with Treplaced by τimmediately yields (3.13). 4. 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:[0,T]×Rn→Rsuch that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ξ→ f(t, ξ) is convex for any t∈[0,T], t→ f(t, ξ)∈W1,1(0,T) for any ξ∈Rn, for any L>0 there exists ˜gL∈L1(0,T) such that |∂tf(t, ξ)|≤˜gL(t) for a.e. t∈[0,T] and all ξ∈BL(0). (4.1) The aim of this section is to prove the following existence result. Theorem 4.1. Let Ω⊂Rnbe a bounded Lipschitz domain and T∈(0,∞). Consider a boundary datum uo∈W1,∞(Ω) such that DuoL∞(Ω,Rn)≤L for a constant L∈(0,∞). Further, assume that the integrand f:[0,T]× Rn→Rsatisfies hypothesis (4.1). Then, there exists a variational solution u∈KL uo(ΩT)to the gradient constrained problem in the sense of Definition 3.1. Further, there holds ∂tu∈L2(ΩT)with the quantitative bound ΩT |∂tu|2dxdt≤4|Ω|sup |ξ|≤L |f(0,ξ)|+˜gLL1(0,T ). We prove Theorem 4.1 via the method of minimizing movements. The proof is divided into five steps.
NoDEA The bounded slope condition for parabolic equations Page 19 of 34 76 4.1. A sequence of minimizers to elliptic variational functionals Fix a step size h:= T mfor some m∈Nand consider times ih,i=0,...,m. For i=0,setu0:= uo∈W1,∞(Ω) with DuoL∞(Ω,Rn)≤L. Further, for i=1,...,m,uiis defined as the minimizer of the elliptic variational functional Fi[v]:=Ω f(ih, Dv)dx+1 2hΩ |v−ui−1|2dx in the class A:= {v∈W1,∞(Ω) : v=uoon∂ΩandDvL∞(Ω,Rn)≤L}.The existence of a minimizer to Fiin this class is ensured by the direct method in the calculus of variations. More precisely, note that A =∅, since uo∈A,and consider a minimizing sequence to Fiin A, i.e. a sequence (ui,j)j∈N⊂Asuch that lim j→∞ Fi[ui,j]= inf v∈A Fi[v]. Further, by definition of Aand Rellich’s theorem there exists a limit map ui∈Aand a (not relabelled) subsequence such that ui,j →uistrongly in L2(Ω) as j→∞, Dui,j Du iweakly in L2(Ω,Rn)asj→∞. Since the functional Fi:W1,2(Ω) →(−∞,∞], Fi[v]:=Fi[v]ifv∈A, ∞else is proper, convex and lower semicontinuous with respect to strong convergence in W1,2(Ω), it is also lower semicontinuous with respect to weak convergence in W1,2(Ω), see [11, Corollary 2.2]. Therefore, we obtain that Fi[ui]= Fi[ui]≤lim inf j→∞ Fi[ui,j] = lim j→∞ Fi[ui,j]= inf v∈A Fi[v]. 4.2. Energy estimates Since ui−1∈Ais an admissible comparison map for the minimizer uiand f fulfills (4.1)3,wehavethat Ω f(ih, Dui)dx+1 2hΩ |ui−ui−1|2dx=Fi[ui] ≤Fi[ui−1] =Ω f((i−1)h, Dui−1)dx+Ω f(ih, Dui−1)−f((i−1)h, Dui−1)dx ≤Ω f((i−1)h, Dui−1)dx+Ω×((i−1)h,ih) |∂tf(t, Dui−1)|dxdt ≤Ω f((i−1)h, Dui−1)dx+|Ω|((i−1)h,ih) |˜gL(t)|dt.
76 Page 20 of 34 L. Schätzler and J. Siltakoski NoDEA Summing up the preceding inequalities from i=1toi=m, we find that m i=1 Ω f(ih, Dui)dxdt+1 2h m i=1 Ω |ui−ui−1|2dx ≤ m i=1 Ω f((i−1)h, Dui−1)dx+|Ω|(0,T ) |˜gL(t)|dt. Subtracting the first term on the left-hand side, we conclude that 1 2h m i=1 Ω |ui−ui−1|2dx≤Ω f(0,Du o)dx−Ω f(T,Dum)dx+|Ω|˜gLL1(0,T ) ≤2|Ω|sup |ξ|≤L |f(0,ξ)|+˜gLL1(0,T ).(4.2) 4.3. The limit map In the following we denote the step size by hmin order to emphasize the dependence on m. First, we join the minimizers uito a map that is piecewise constant with respect to time. More precisely, we define u(m):Ω×(−hm,T]→ Rby u(m)(t):=uifor t∈((i−1)hm,ih m],i=0,...,m. Observe that the sequence u(m)m∈Nis bounded in L∞(ΩT), since u(m)L∞(ΩT)= maxi=0,...,m uiL∞(Ω),ui∈Afor all i=0,...,m and Ais equibounded. Further, we know that Du(m)L∞(ΩT,Rn)= maxi=0,...,m DuiL∞(Ω,Rn)≤Lfor any m∈N. Therefore, there exists a subsequence K⊂Nand a limit map u∈L∞(ΩT) such that DuL∞(ΩT,Rn)≤L,u=uo on ∂Ω×(0,T)and ⎧ ⎨ ⎩ u(m)∗ uweakly ∗inL∞(ΩT)asKm→∞, u(m)(t)→u(t) uniformly as Km→∞for eacht∈[0,T], Du(m)∗ Duweakly ∗inL∞(ΩT,Rn)asKm→∞. (4.3) In order to prove that uhas a time derivative, we consider the linear interpolation of minimizers ˜u(m):Ω×(−hm,T]→Rgiven by ˜u(m)(t):=uofor t∈(−hm,0] and ˜u(m)(t):=i−t hmui−1+1−i+t hmuifor t∈((i−1)hm,ih m],i=1,...,m. Similar arguments as above ensure that ˜u(m)m∈Nis bounded in L∞(ΩT)and that D˜u(m)L∞(ΩT,Rn)≤Lfor any m∈N. Moreover, by the energy bound (4.2) we obtain that ΩT |∂t˜u(m)|2dxdt= m i=1 Ω×((i−1)hm,ihm] 1 h2 m|ui−ui−1|2dxdt =1 hm m i=1 Ω |ui−ui−1|2dx ≤4|Ω|sup |ξ|≤L |f(0,ξ)|+˜gLL1(0,T ).(4.4)
NoDEA The bounded slope condition for parabolic equations Page 21 of 34 76 Hence, ˜u(m)m∈Nis bounded in W1,2(ΩT). By Rellich’s theorem we conclude that there exists a subsequence still labelled Kand a limit map ˜u∈L∞(ΩT) with D˜uL∞(ΩT,Rn)≤L,˜u=uoon ∂Ω×(0,T)and∂t˜u∈L2(ΩT) such that ˜u(m)→ustrongly in L2(ΩT)asKm→∞, ∂t˜u(m)∂ t˜uweakly in L2(ΩT)asKm→∞.(4.5) Note that ∂t˜u∈L2(ΩT) in particular implies that ˜u∈C0; 1 2([0,T]; L2(Ω)) and therefore ˜uis contained in the class of functions KL uo(ΩT). Next, since u(m)−˜u(m)(t)≤|ui−ui−1|for t∈((i−1)hm,ih m], i=1,...,m, we infer from (4.2) that ΩTu(m)−˜u(m)2dxdt≤hm m i=1 Ω |ui−ui−1|2dx ≤4|Ω|sup |ξ|≤L |f(0,ξ)|+˜gLL1(0,T )h2 m. Together with (4.5)1this implies that u(m)→˜ustrongly in L2(ΩT)asK m→∞and thus in particular that u=˜u∈KL uo(ΩT) with ∂tu∈L2(ΩT). Finally, by lower semicontinuity with respect to weak convergence, (4.4) gives us the claimed bound ΩT |∂tu|2dxdt≤4|Ω|sup |ξ|≤L |f(0,ξ)|+˜gLL1(0,T ). 4.4. Minimizing property of the approximations First, define piecewise constant approximations of the integrand by f(m)(t, ξ):=f(ih, ξ)fort∈((i−1)hm,ih m],i=0,...,m. We claim that u(m)is a minimizer of the functional F(m)[v]:=ΩT f(m)(t, Dv)dxdt+1 2hmΩT |v(t)−u(m)(t−hm)|2dxdt in the class of functions AT:= {v∈L∞(ΩT):DuL∞(ΩT,Rn)≤Land u=uoon ∂Ω×(0,T)}. Indeed, consider an arbitrary map v∈A T. Since v(t)∈Afor a.e. t∈[0,T], by the minimizing property of uiwith respect to Fiin the class Awe find that F(m)u(m)= m i=1 ((i−1)hm,ihm] Fi[ui]dt≤ m i=1 ((i−1)hm,ihm] Fi[v(t)] dt=F(m)[v]. A straightforward computation shows that this is equivalent to ΩT f(m)t, Du(m)dxdt ≤ΩT f(m)(t, Dv)dxdt +1 hmΩT 1 2v−u(m)2+v−u(m)u(m)−u(m)(t−hm)dxdt
76 Page 22 of 34 L. Schätzler and J. Siltakoski NoDEA for any v∈A T. Choosing the convex combination u(m)+sv−u(m)∈A T with s∈(0,1) as comparison map and using the convexity of ξ→ f(t, ξ)for all t∈[0,T], we obtain that ΩT f(m)t, Du(m)dxdt ≤(1 −s)ΩT f(m)t, Du(m)dxdt+sΩT f(m)(t, Dv)dxdt +1 hmΩT s2 2v−u(m)2+sv−u(m)u(m)−u(m)(t−hm)dxdt. Reabsorbing the first term on the right-hand side into the left-hand side, dividing the resulting inequality by sand taking the limit s↓0 gives us that ΩT f(m)t, Du(m)dxdt ≤ΩT f(m)(t, Dv)dxdt+1 hmΩTv−u(m)u(m)−u(m)(t−hm)dxdt. Next, assume without loss of generality that v(0) ∈L∞(Ω), extend vto (−hm,0] by v(0) and note that v−u(m)u(m)−u(m)(t−hm) =v−u(m)v−v(t−hm)+1 2v(t−hm)−u(m)(t−hm)2−1 2v−u(m)2 −1 2v−v(t−hm)−u(m)+u(m)(t−hm)2 ≤v−u(m)v−v(t−hm)+1 2v(t−hm)−u(m)(t−hm)2−1 2v−u(m)2. Inserting this into the preceding inequality and recalling that v(t)=v(0) for t∈(−hm,0], we infer ΩT f(m)t, Du(m)dxdt ≤ΩT f(m)(t, Dv)dxdt+1 hmΩTv−u(m)v−v(t−hm)dxdt (4.6) +1 2hmΩTv(t−hm)−u(m)(t−hm)2−v−u(m)2dxdt =ΩT f(m)(t, Dv)dxdt+1 hmΩTv−u(m)v−v(t−hm)dxdt +1 2Ω (v−uo)2dx−1 2hmΩ×(T−hm,T ]v−u(m)(T)2dxdt. 4.5. Variational inequality for the limit map We fix an arbitrary map v∈KL uo(ΩT) with ∂tv∈L2(ΩT). Thus, in particular we have that v∈A T,sovis an admissible comparison map in (4.6). Our goal is to pass to the limit Km→∞in (4.6) in order to deduce the variational
NoDEA The bounded slope condition for parabolic equations Page 23 of 34 76 inequality (3.1)foru. To this end, we consider the terms separately. First, we write the first term on the left-hand side of (4.6)as ΩT f(m)t, Du(m)dxdt =ΩT ft, Du(m)dxdt+ΩT f(m)t, Du(m)−ft, Du(m)dxdt. By Lemma 2.4 and (4.3)3, we obtain that ΩT f(t, Du)dxdt≤lim inf Km→∞ ΩT ft, Du(m)dxdt. Further, since Du(m) L∞(ΩT,Rn)≤Lfor all m∈Nand ffulfills (4.1)3,we estimate ΩT f(m)t, Du(m)−ft, Du(m)dt ≤ m i=1 Ω×((i−1)hm,ihm]fihm,Du (m)−ft, Du(m)dxdt ≤ m i=1 Ω×((i−1)hm,ihm]((i−1)hm,ihm]∂tfs, Du(m)(t)dsdxdt ≤|Ω|hm m i=1 ((i−1)hm,ihm] ˜gL(s)ds =|Ω|˜gLL1(0,T )hm. Therefore, this term vanishes in the limit m→∞. Joining the preceding estimates, we conclude that ΩT f(t, Du)dxdt≤lim inf Km→∞ ΩT f(m)t, Du(m)dxdt. (4.7) 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 ΩT f(t, Dv)dxdt= lim m→∞ ΩT f(m)(t, Dv)dxdt. (4.8) Next, since 1 hm(v(t)−v(t−hm)) →∂tvstrongly in L2(ΩT)andu(m)u weakly in L2(ΩT)asKm→∞by (4.3)1, we have that ΩT ∂tv(v−u)dxdt= lim Km→∞ 1 hmΩTv−u(m)v−v(t−hm)dxdt. (4.9) Finally, by the fact that v∈C0([0,T]; L2(Ω)) and by (4.3)2, we obtain that 1 2(v−u)(T)2 L2(Ω) = lim Km→∞ 1 2hmΩ×(T−hm,T ]v−u(m)(T)2dxdt. (4.10)
76 Page 24 of 34 L. Schätzler and J. Siltakoski NoDEA Collecting the assertions (4.7)–(4.10) yields ΩT f(t, Du)dxdt≤ΩT f(t, Dv)dxdt+ΩT ∂tv(v−u)dxdt +1 2v(0) −uo2 L2(Ω) −1 2(v−u)(T)2 L2(Ω). Since v∈KL uo(ΩT) with ∂tv∈L2(ΩT) was arbitrary, we have shown that u∈KL uo(ΩT) is the desired variational solution. 5. 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 7.1]. It states that affine functions independent of time are variational solutions to (1.1) with respect to their own initial and lateral boundary values. Lemma 5.1. Let Ωbe open and bounded. Assume that f:[0,T]×Rn→R satisfies (1.2).Letw(x, t):=a+ξ·xwith constants a∈Rand ξ∈Rnbe an affine function independent of time. Then wis a variational solution in the sense of Definition 1.1 in K∞ w(ΩT). With the preceding lemma at hand, we are able to prove the following. Theorem 5.2. Let T∈(0,∞), assume that Ω⊂Rnis open, bounded and convex, and that the integrand f:[0,T]×Rn→Rsatisfies (4.1). Consider uo∈W1,∞(Ω) such that DuoL∞(Ω,Rn)≤Qand suppose that uo|∂Ωsatisfies the bounded slope condition with the same parameter Q. Then there exists a variational solution u∈K∞ uo(ΩT)to (1.1)in the sense of Definition 1.1. Further, we have the quantitative bound DuL∞(ΩT,Rn)≤Q. (5.1) Proof. Let L>Q. By Theorem 4.1 there exists a variational solution u∈ KL uo(ΩT) with ∂tu∈L2(ΩT) 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 uis in fact already a solution to the unconstrained problem. Fix xo∈∂Ω and denote by w± xothe affine functions from Lemma 2.2.In particular we have w− xo≤uo≤w+ xo. Since by Lemma 5.1 the functions w− xo and w+ xoare variational solutions, it follows from the comparison principle in Theorem 3.5 that w+ xo(x)≤u(x, t)≤w− xo(x) for all (x, t)∈ΩT. Consequently, there holds |u(x, t)−uo(xo)|≤Q|x−xo|for all (x, t)∈ΩT.
NoDEA The bounded slope condition for parabolic equations Page 31 of 34 76 Passing to the limit δ↓0, the preceding inequality implies that |uη(τ1)−uη(τ2)|≤τ2 τ1 − Br∩Ω |F·Dη|dxdt ≤(τ2−τ1)DηL∞(Ω,Rn)sup (x,t)∈Qr |F(x, t)| =2c(n, Ω)rsup (x,t)∈Qr |Dξf(t, Du(x, t))| holds true for almost every τ1,τ 2∈(t1,t 2). In the last inequality, we used that τ2−τ1≤t2−t1≤2r2.Thus I1≤c(n, Ω)r2sup (x,t)∈Qr |Dξf(t, Du(x, t))|2.(7.6) Inequality (7.3) now follows by combining the estimates of I1,I 2and I3. Acknowledgements Jarkko Siltakoski was funded by the Magnus Ehrnrooth foundation. Author contributions The authors contributed equally to the article. Funding Open Access funding provided by University of Jyv¨askyl¨a (JYU). Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References [1] Bebendorf, M.: A note on the Poincar´e inequality for convex domains. Z. Anal. Anwendungen 22(4), 751–756 (2003)
76 Page 32 of 34 L. Schätzler and J. Siltakoski NoDEA [2] Bousquet, P.: On the lower bounded slope condition. J. Convex Anal. 14(1), 119–136 (2007) [3] Bousquet, P.: Boundary continuity of solutions to a basic problem in calculus of variations. Adv. Calc. Var. 3(1), 1–27 (2010) [4] Bousquet, P., Brasco, L.: Global Lipschitz continuity for minima of degenerate problems. Math. Ann. 366(3–4), 1403–1450 (2016) [5] B¨ogelein, V., Duzaar, F., Marcellini, P.: Parabolic systems with p, q-growth: a variational approach. Arch. Ration. Mech. Anal. 210(1), 219–267 (2013) [6] B¨ogelein, V., Duzaar, F., Mingione, G.: The regularity of general parabolic systems with degenerate diffusion. Mem. Am. Math. Soc. 221(1041), 1–143 (2013) [7] B¨ogelein, V., Duzaar, F., Marcellini, P., Signoriello, S.: Parabolic equations and the bounded slope condition. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 34(2), 355–379 (2017) [8] B¨ogelein, V., Stanin, T.: The one-sided bounded slope condition in evolution problems. Ann. Mat. Pura Appl. (4) 199(2), 573–587 (2020) [9] Da Prato, G.: Spazi L(p,ϑ)(Ω,δ) e loro proprieta. Ann. Math. Pura Appl. 69(4), 383–392 (1965) [10] Don, S., Lussardi, L., Pinamonti, A., Treu, G.: Lipschitz minimizers for a class of integral functionals under the bounded slope condition. Nonlinear Anal. 216, 112689 (2016) [11] Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics, Philadelphia (1999) [12] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions (Revised Version). Studies in Advanced Mathematics, CRC Press, Boca Raton (2015) [13] Giannetti, F., Treu, G.: On the Lipschitz regularity for minima of functionals depending on x,u,and∇uunder the bounded slope condition. SIAM J. Control Optim. 60(3), 1347–1364 (2022) [14] Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2003) [15] Haar, A.: ¨ Uber das Plateausche Problem. Math. Ann. 97(1), 124–158 (1927) [16] Hardt, R., Zhou, X.: An evolution problem for linear growth functionals. Commun. Partial Differ. Equ. 19(11&12), 1879–1907 (1994) [17] Hartman, P., Nirenberg, L.: On spherical image maps whose Jacobians do not change sign. Am. J. Math. 115, 271–310 (1966) [18] Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966)
NoDEA The bounded slope condition for parabolic equations Page 33 of 34 76 [19] Kinnunen, J., Lindqvist, P.: Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation. Ann. Mat. Pura Appl. (4) 185(3), 411–435 (2006) [20] Lichnewsky, A., Temam, R.: Pseudosolutions of the time-dependent minimal surface problem. J. Differ. Equ. 30(3), 340–364 (1978) [21] Marcellini, P.: A variational approach to parabolic equations under general p, qgrowth conditions. Nonlinear Anal. 194, 111456–17 (2020) [22] Mariconda, C., Treu, G.: Existence and Lipschitz regularity for minima. Proc. Am. Math. Soc. 130(2), 395–404 (2002) [23] Mariconda, C., Treu, G.: Lipschitz regularity for minima without strict convexity of the Lagrangian. J. Differ. Equ. 243(2), 388–413 (2007) [24] Mariconda, C., Treu, G.: A Haar-Rado type theorem for minimizers in Sobolev spaces. ESAIM Control Optim. Calc. Var. 17(4), 1133–1143 (2011) [25] Mingione, G., Rˇadulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501(1), 125197–41 (2021) [26] Miranda, M.: Un teorema di esistenza e unicit`a per il problema dell’area minima in n variabili. Ann. Sc. Norm. Super. Pisa 19(3), 233–249 (1965) [27] Rainer, R., Siltakoski, J., Stanin, T.: An evolutionary Haar-Rado theorem. Manuscr. Math. 168(1–2), 65–88 (2022) [28] Sch¨atzler, L.: Existence of variational solutions for time dependent integrands via minimizing movements. Analysis (Berlin) 37(4), 199–222 (2017) [29] Stampacchia, G.: On some regular multiple integral problems in the calculus of variations. Commun. Pure Appl. Math. 16, 383–421 (1963) [30] Stanin, T.: Global continuity of variational solutions weakening the one-sided bounded slope condition. Forum Math. 33(5), 1237–1260 (2021) Leah Sch¨atzler Fachbereich Mathematik Paris-Lodron-Universit¨at Salzburg Hellbrunner Straße 34 5020 Salzburg Austria e-mail: leahanna.sc[email protected]
76 Page 34 of 34 L. Schätzler and J. Siltakoski NoDEA Jarkko Siltakoski Department of Mathematics and Statistics University of Jyv¨askyl¨a P.O.Box 3540014 Jyv¨askyl¨a Finland e-mail: jarkko.j.m.siltakoski@jyu.fi Received: 8 September 2022. Revised: 8 January 2023. Accepted: 12 July 2023.