An evolutionary Haar-Rado type theorem
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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/ An evolutionary Haar-Rado type theorem © The Author(s) 2021 Published version Rainer, Rudolf; Siltakoski, Jarkko; Stanin, Thomas Rainer, R., Siltakoski, J., & Stanin, T. (2022). An evolutionary Haar-Rado type theorem. Manuscripta Mathematica, 168(1-2), 65-88. https://doi.org/10.1007/s00229-021-01293-8 2022
manuscripta math. © The Author(s) 2021 Rudolf Rainer ·Jarkko Siltakoski ·Thomas Stanin An evolutionary Haar-Rado type theorem Received: 18 August 2020 / Accepted: 13 March 2021 Abstract. In this paper, we study variational solutions to parabolic equations of the type ∂tu−divx(Dξf(Du))+Dug(x,u)=0, where uattains time-independent boundary values u0on the parabolic boundary and f,gfulfill convexity assumptions. We establish a HaarRado type theorem: If the boundary values u0admit a modulus of continuity ωand the estimate |u(x,t)−u0(γ )|≤ω(|x−γ|)holds, then uadmits the same modulus of continuity in the spatial variable. 1. Introduction In this article, we are concerned with establishing a Haar-Rado type theorem for the following kind of Cauchy-Dirichlet problem: ∂tu−divx(Dξf(Du)) +Dug(x,u)=0inT, u=u0on ∂PT,(1.1) for a time-independent boundary datum u0:→R.ForT>0, the set T:=× (0,T)is a space-time cylinder with a bounded domain ⊂Rn,n≥2. As usual, by ∂PT:= ( ×{0})∪(∂ ×(0,T)) we denote the parabolic boundary of T, use the operator symbol ∂tto express a derivative with respect to the time variable, and write divxfor the spatial divergence operator. The classical Haar-Rado theorem (see e.g. [15, Prop. 2.11] and the references therein) was originally formulated in an elliptic setting, which is concerned with the minimization problem min f(Du)dx:u∈Lipφ(),(1.2) Rudolf Rainer (B): Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria e-mail: [email protected] Jarkko Siltakoski: Department of Mathematics and Statistics, University of Jyväskylä, P.O.Box 35, FIN-40014 Jyväskylä, Finland e-mail: jarkko.siltak[email protected] Thomas Stanin: Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria e-mail: [email protected] Mathematics Subject Classification: 35A15 ·35B65 ·35K55 ·49J40 https://doi.org/10.1007/s00229-021-01293-8
R. Rainer et al. where fis strictly convex and Lipφ() denotes the space of Lipschitz functions having boundary values equal to the given Lipschitz function φon the boundary ∂ of an open and bounded set ⊂Rn. The theorem then states that a minimizer ufulfills the property sup x,y∈ |u(x)−u(y)| |x−y|=sup x∈,γ∈∂ |u(x)−φ(γ)| |x−γ|.(1.3) As seen in [15, Prop. 11.41], this can be used to obtain global gradient estimates on from mere boundary estimates on ∂ and it also plays a fundamental role in the existence proof (cf. [16, Lemma 1.3]) for Lipschitz minimizers to (1.2) whenever the boundary datum φfulfills a barrier condition, such as the quite restrictive bounded slope condition (see Definition 2.7 and [19,27] for some discussion). This condition has a long tradition within a sub-field of Calculus of Variations, which is now known as Hilbert-Haar theory and is based on papers from Haar [17], Hartman and Nirenberg [20], Stampacchia [28], Miranda [27], and Hartman and Stampacchia [21]. In recent years, many authors were investigating questions considering existence and regularity of minimizers to a more general class of problems of the form min f(Du)+g(x,u)dx:u∈W1,p φ(),(1.4) where – for a suitable domain ⊂Rnand p≥1–W1,p φ() denotes the space of Sobolev functions u∈W1,p() having trace equal to φon ∂. For a selection of results covering the case g=0, we refer the interested reader to [6,10–12,24,25], as well as to [7–9,14,23] for some results about the general case g= 0. In this context, the bounded slope condition was also considerably weakened in a few ways, one of them being a one-sided bounded slope condition (see [12]). Following this approach, in [26] Mariconda and Treu reformulated and generalized the classical Haar-Rado theorem (1.3) to the new setting (1.4)forp=1, even allowing φto admit a general modulus of continuity ω. More precisely, they showed that if, among other assumptions, |u(x)−φ(γ)|≤ω(|x−γ|)for a.e. x∈and for all γ∈∂, this estimate can be extended to the interior of , i.e. uhas an ω-continuous representative. Note that this is a reformulation of (1.3) in the case that ωis a Lipschitz-modulus. Our main goal in this paper is to extend this statement to the parabolic setting, in terms of variational solutions. For a given variational solution uto (1.1), attaining ω-continuous and time-independent boundary data u0on the parabolic boundary, we show that if |u(x,t)−u0(x0)|≤ω(|x−x0|)for a.e. (x,t)∈Tand for all x0∈∂, then, as before, this estimate can be extended slice-wise to the interior of the spacetime cylinder T, provided that the intersection of with a shifted version of itself has a Lipschitz boundary.
An evolutionary Haar-Rado type theorem Wepointoutagainthatwetreattime-independentboundarydata.Fromourpoint of view the investigation of boundary data which depend on time, i.e. u0=u0(x,t) requires further investigation and could be an interesting subject for future studies. Following the tradition of Hilbert-Haar theory, we will not suppose any growth conditions from above on for g. For this reason we cannot use the notion of weak solutions and instead take the so called variational approach in the spirit of Lichnewsky and Temam [22]. This approach to evolutionary equations has become increasingly popular in the last several years and the existence of variational solutions has been obtained to rather general equations, see for example [1–3]. Regularity of variational solutions was studied by Bögelein, Duzaar, Marcellini and Signoriello [4], who obtained the spatial Lipschitz continuity of solutions to (1.1) when g≡0 and u0satisfies the bounded slope condition. Older results regarding regularity proofs of weak solutions via the bounded slope condition in an evolutionary setting with functionals of linear growth can for example be found in Hardt and Zhou [18, Chapter 4]. Moreover, recently Bögelein and Stanin [5] proved local Lipschitz continuity in spacetime of variational solutions under an evolutionary variant of the previously mentioned one-sided bounded slope condition. Our article will be structured as follows: In Sect. 2we present the setting and our main result. Several lemmata, which are useful for our purposes, are collected in Sect. 3. Then we proceed with the proof of our main result, which can be summarized as follows: We compare the variational solution uwith the shifted variational solution, defined on a shifted space-time cylinder. Topic of Sect. 4is the proof that this shift retains the property of being a variational superor sub-solution. In the main step, which is carried out in Sect. 5, we can thus apply a comparison principle on the intersection of the original cylinders. Finally, in Sect. 6we apply our main result to obtain the spatial Lipschitz continuity of variational solutions in the setting of the bounded slope condition. 2. Definitions and main result Throughout this article, we denote by T:=×(0,T)a spacetime cylinder with T>0 and a bounded Lipschitz domain ⊂Rn. For a function v∈L1(T)≡ L1([0,T]; L1()), we will often adopt the abbreviation v(t):=v(·,t)to denote the evaluation of the function vat time t∈[0,T], which is to be understood in a suitable sense. Weconsiderthe Cauchy-Dirichlet problem (1.1) withtwovariational integrands f,g, which we impose different conditions on. For the (only gradient-dependent) variational integrand f:Rn→Rwe require that fis convex, fis p-coercive, i.e. f(ξ) ≥μ|ξ|p+νfor all ξ∈Rn,some μ>0andν∈R,(2.1)
R. Rainer et al. with an exponent p>1. For the (xand u-dependent) variational integrand g:×R→Rwe require that ⎧ ⎨ ⎩ x→ g(x,u)is measurable for all u∈R, u→ g(x,u)is convex for almost every x∈, g(x,u)≥−k(x)(1+|u|)for a.e. x∈and every u∈R, ⎫ ⎬ ⎭(2.2) where k∈Lp(;R≥0)with p:= p p−1denoting the conjugate Hölder exponent of p. Due to convexity, the right derivative of gwith respect to the second variable exists, is monotonically non-decreasing and we denote it by g+ u. Furthermore, we impose the following conditions on the boundary datum u0: u0∈W1,p() ∩L2(), f(Du0)+g(x,u0)dx<∞(2.3) As mentioned, we do not demand any growth conditions from above on neither f nor g, and therefore we cannot use the notion of weak solutions in this context. Instead, we use variational solutions – a notion going back to the article [22] of Lichnewsky and Temam – whose existence to (1.1) is guaranteed due to our structural assumptions on f,gand u0(cf. Theorem 2.2). If fand gare suitably regular, then variational solutions coincide with weak solutions [2]. In order to define variational solutions, we first restate the definition of typical function spaces in the parabolic setting: The parabolic Sobolev space Lp([0,T]; W1,p()) is the set of all measurable functions u:T→Rsuch that x→ u(x,t)is in W1,p() for a.e. t∈(0,T)and further T |u|p+|Du|pdz<∞.(2.4) Similarly, for v∈Lp([0,T]; W1,p()) we denote by v+Lp([0,T]; W1,p 0()) the set of all measurable functions such that x→ u(x,t)−v(x,t)belongs to W1,p 0() for almost every t∈(0,T)and (2.4) is fulfilled. Since we will also need weaker versions of the mentioned variational solutions later on, we define them in the following lines as well: Definition 2.1. (Variational (sub-/super-) solution) We call a function u∈Lp([0,T]; W1,p()) ∩L2(T)with ∂tu∈L2(T) avariational sub-solution (respectively a variational super-solution) on Tif T f(Du)+g(x,u)dz<∞ and the inequality T f(Du)+g(x,u)dz≤T ∂tv(v −u)+f(Dv) +g(x,v)dz −1 2 (v −u)2(x,·)dxT 0 (2.5)
An evolutionary Haar-Rado type theorem holds true for any v∈u+Lp([0,T]; W1,p 0()) ∩L2(T)with v≤ualmost everywhere in T(resp. v≥ualmost everywhere in T), additionally satisfying ∂tv∈L2(T). Furthermore, we call uavariational solution if it is both variational suband super-solution. In this case, any v∈u+Lp([0,T]; W1,p 0()) ∩L2(T) with ∂tv∈L2(T)is an admissible test function. Finally, such uis called a variational solution to the Cauchy-Dirichlet problem (1.1)ifu(x,t)=u0(x)on the lateral boundary ∂ ×(0,T)in the sense of traces and u(·,0)=u0(·). Note that for a variational solution one usually demands that u∈Lp([0,T]; W1,p()) ∩C0([0,T]; L2()), where C0([0,T]; L2()) denotes the Bochner space of continuous functions u: [0,T]→L2() such that max t∈[0,T] |u(x,t)|2dx<∞. However, it is worth citing the existence and regularity result obtained in [2, Theorem 1.2]. It ensures existence of variational (suband super-)solutions and also includes important regularity properties. For the sake of completeness, we include thestatement, adapted toour setting, whichis a special case of the original Theorem. Theorem 2.2. Let f satisfy the conditions given in (2.1), g the assumptions in (2.2) and suppose u0to fulfill the constraints in (2.3). Then, there exists a variational solution (and thus also a subresp. super-solution) u ∈Lp([0,T]; W1,p()) ∩ C0([0,T]; L2()) to (1.1), that further fulfils u ∈C0,1/2[0,T]; L2()and admits a weak time derivative ∂tu∈L2(T). The property ∂tu∈L2(T)will be an important technical tool in the proofs. Thus,weimposethisconditioninthedefinitionabove. Further note that the property u∈C0([0,T]; L2()) follows from ∂tu∈L2(T)by using a standard mollifier argument. Definition 2.3. (Modulus of continuity) A continuous and increasing function ω: [0,∞)→[0,∞)with ω(0)=0 is called a modulus of continuity. For a set X⊂ Rn, a function φ:X→Ris called ω-continuous if |φ(x)−φ(y)|≤ω(|x−y|) for all x,y∈X. Later, wewill need thefactthat shifting leavesthe property of being a variational super-solution invariant (cf. Theorem 4.2). Due to the x-dependency of the function g, mere convexity is not enough to obtain this. We need the following stronger assumption: For h∈Rn,x∈Rand a modulus of continuity ω, we demand v≥u+ω(|h|)⇒g+ u(x−h,v)≥g+ u(x,u)∀x∈Rn,∀u,v ∈R.(Hh,ω) If gdoes not depend on xand is convex, then this property is fulfilled. For another sufficient condition, see [26, Prop. 4.1].
R. Rainer et al. We define the shifted set h:= {x+h:x∈}and denote the corresponding shifted spacetime cylinder by h,T:= h×(0,T). Additionally, for a function u:T→R, we define the associated shifted function uh:h,T→Rvia uh(x,t):= u(x−h,t). (2.6) Now we are able to formulate our main result, which is the following: Theorem 2.4. (Haar-Rado type) Suppose that the conditions (2.1)-(2.3)and (Hh,ω)hold. Further suppose u to be a variational solution to (1.1)and assume u0 to be ω-continuous on with a given modulus of continuity ω. Additionally, let ∂( ∩h)be Lipschitz for all h ∈Rnand suppose that the estimate |u(x,t)−u0(x0)|≤ω(|x−x0|)(2.7) holds true for any x0∈∂ and almost every (x,t)∈T. Then, u satisfies |u(y,t)−u(x,t)|≤ω(|y−x|), whenever (x,t)and (y,t)∈Tare Lebesgue points of u. Note that the requirement that has a Lipschitz boundary does not imply that ∂( ∩h)is Lipschitz. The reason for imposing the conditions on the boundary is to ensure the existence of the trace operator. For further details, see [26, Section 3.2]. Corollary 2.5. Suppose that the conditions (2.1)-(2.3)and (Hh,ω)hold. Let ∂(∩ h)be Lipschitz. Suppose that u is a variational solution to (1.1)on Twith initial and boundary data u0. Suppose that u0is ω-continuous in . Suppose that there exist l1,l2∈W1,p u0() that are ω-continuous on and l1≤u≤l2in T. Then we have |u(y,t)−u(x,t)|≤ω(|y−x|) whenever (x,t),(y,t)∈Tare Lebesgue points of u. A straight-forward situation in which ∂( ∩h)is Lipschitz for any h>0, is if is bounded and convex. We thus formulate the following corollary, with the often used Hölder-continuity: Corollary 2.6. Suppose that the conditions (2.1)-(2.3)and (Hh,ω)hold. Let be a bounded and convex domain. Suppose that u is a variational solution to (1.1)in T with initial and boundary data u0.Forα∈(0,1)let u0be α-Hölder-continuous on . Assume that |u(x,t)−u0(x0)|≤|x−x0|α holds true for any x0∈∂ and almost every (x,t)∈T. Then, u satisfies |u(y,t)−u(x,t)|≤|y−x|α, whenever (x,t)and (y,t)∈Tare Lebesgue points of u.
An evolutionary Haar-Rado type theorem As an application of our main result, we obtain the spatial Lipschitz continuity of variational solutions to (1.1) under certain assumptions. In particular, we suppose the so called bounded slope condition on the initial and boundary data u0. Definition 2.7. (Bounded slope condition) A function u0:∂ →Rsatisfies the bounded slope condition with constant Q >0 if for any x0∈∂ there exist affine functionsa1,a2:Rn→Rsuchthat|Da1|,|Da2|<Q,a1(x0)=u0(x0)=a2(x0) and a1≤u0≤a2on ∂. The bounded slope condition forces u0to be an affine function on flat parts of ∂ and to be convex unless u0is an affine function. If is a uniformly convex and bounded C2domain, then u0|∂ satisfies the bounded slope condition whenever u0∈C2(Rn),see[16,27] for more details. Note that Corollary 2.5 cannot be immediately applied in the setting of the bounded slope condition, since in general the non-linearity gprevents affine functions from being variational subor super-solutions and thus from being admissible comparison functions. It is hence not possible, in this manner, to obtain an inequality of the form a1≤u≤a2in Twith a1,a2affine functions. One way to deal with this problem is to assume that the Lagrangian is uniformly convex in a suitable sense. The strategy is then to bend the affine functions in space so that the divergence term dominates the lower order term gin the Euler-Lagrange equation, causing the bent functions to be suitable barriers. This kind of ideas were previously used in the time-independent setting, see for example [28] and [8]. Definition 2.8. Let :[0,∞)→[0,∞)be a continuous function such that lim s→∞ s(s)=+∞. We say that f∈C2(Rn)is -uniformly convex if D2f(η)ξ ·ξ=tr(D2f(η)ξ ⊗ξ) ≥(|η|)|ξ|2 for all ξ,η ∈Rn.Hereξ⊗ξdenotes the n×nmatrix whose (i,j)entry is ξiξj. Note that this condition can be interpreted as a generalization of a coercivity conditionimposedonthematrix D2f(η) orthebilinearform(ζ, ξ) → D2f(η)ζ ·ξ, respectively. In particular, for a constant function , this condition reduces to a usual coercivity condition well-known from elliptic PDE theory and Definition 2.8 demands fto be a strongly convex function. Proposition 2.9. Suppose that the conditions (2.1)-(2.3)hold. Further suppose that is convex, (Hh,ω)holds with a Lipschitz modulus of continuity and that f∈C2(Rn)is -uniformly convex. Suppose moreover that u0is Lipschitz in and u0|∂ satisfies the bounded slope condition with Q >0. Then there is L >0 such that a variational solution u to the Cauchy-Dirichlet problem (1.1)satisfies |u(x,t)−u(y,t)|≤L|x−y| whenever (x,t), (y,t)∈Tare Lebesgue points of u.
R. Rainer et al. 3. Preliminary and auxiliary results In this section, we state some important results which will be used in our proofs in the later chapters. We start by proving both spatial and time localization principles. For the convenience of the reader, we will also give the proofs. Lemma 3.1. (Spatiallocalizationprinciple)Let ˜ ⊂⊂Rnbe bounded Lipschitz domains. Suppose u :T→Rto be a variational sub-solution (resp. a variational super-solution) on T, Then, u|˜ T:˜ T→Ris a variational sub-solution (resp. a variational super-solution) on ˜ T. Proof. We choose a function w∈u˜ T+Lp([0,T]; W1,p 0(˜ )) ∩L2(T)with w≤u|˜ Ta.e. on ˜ T(resp. w≥u|˜ Ta.e. on ˜ T) and ∂tw∈L2(˜ T). Observe that v:= win ˜ T, uin ( \˜ )T, is a valid test function in the inequality (2.5) of the variational sub-solution u(resp. the variational super-solution u)onT. Inserting this test function immediately leads to ˜ T f(Du)+g(x,u)dz≤˜ T ∂tw(w −u)+f(Dw) +g(x,w)dz −1 2˜ (w −u)2(x,·)dxT 0. Since wwas chosen arbitrarily at the beginning, the last inequality proves that u|˜ Tis a variational sub-solution (resp. a variational super-solution) on ˜ , which concludes the proof. Lemma 3.2. (Temporal localization principle) Let ⊂Rnbe a bounded Lipschitz domain, 0<t1<t2<T , and define t1,t2:=×(t1,t2). Suppose u :T→R to be a variational sub-solution (resp. a variational super-solution) on T. Then, the restriction u|t1,t2is a variational sub-solution (resp. super-solution) on t1,t2. Proof. In the following, we will use the convenient abbreviation ut1,t2:=u|∂Pt1,t2. We choose a function w∈u+Lp([t1,t2]; W1,p 0())∩L2(t1,t2)with w≤u|t1,t2 a.e. in t1,t2(resp. w≥u|t1,t2a.e. in t1,t2), ∂tw∈L2(t1,t2). Additionally, we define for ϑ∈(0,t2−t1 2)a cutoff function ζϑ∈W1,∞([0,T])with respect to time via ζϑ(t):=t−t1 ϑ 1[t1,t1+ϑ)(t)+1[t1+ϑ,t2−ϑ](t)+t2−t ϑ 1(t2−ϑ,t2](t) and set v:=ζϑw+(1−ζϑ)u.
An evolutionary Haar-Rado type theorem where g+ udenotes the right-derivative of gwith respect to the second variable. We insert c=ω(|h|), which allows us to use the assumption (Hh,ω)to obtain ψ +(y)=g+ u(x−h,y)≥g+ u(x,y−c)=φ +(y) in [u(x)+c,˜v(x−h)]. Thus, integration over this interval yields ψ(˜v(x−h)) −ψ(u(x)+c)≥φ(˜v(x−h)) −φ(u(x)+c)), which is exactly (4.2). Thus ˜ufulfills the variational inequality and is a variational super-solution. To obtain the respective result for sub-solutions, the calculations are similar. In the assumption (Hh,ω) one can then replace xwith x+hand set v=y+c,u=cto come to the analogous conclusion. 5. Proof of the main result This section is dedicated to proving the main result (Theorem 2.4) of this article. We begin with a technical lemma which says that the Haar-Rado type condition (2.7)impliesthat uh(·,t)−ω(|h|)≤u(·,t)on ∂( ∩)h in the sense of traces for any h∈Rn. Since the shifted function at the left-hand side is still a variational sub-solution by Theorem 4.2, while the boundedness will be shown in Lemma 5.2, it follows from the comparison principle that the inequality holds almost everywhere in ∩h. Writing h=x−ythen essentially yields u(x,t)−u(y,t)≤ω(|x−y|)whenever (x,t)and (y,t)are Lebesgue points of u. Lemma 5.1. Let ∂( ∩h)be Lipschitz. Suppose that u0∈W1,p() is ωcontinuous on and let u∈u0+Lp([0,T]; W1,p 0()) ∩L2(T)with ∂tu∈L2(T). Suppose moreover that |u(x,t)−u0(γ )|≤ω(|x−γ|)for a.e. (x,t)∈Tand γ∈∂. (5.1) Then uh(·,t)−ω(|h|)≤u(·,t)on ∂( ∩h) in the sense of traces for almost all t ∈(0,T). Proof. For the properties of the trace that we use in this proof, we once again refer to [26, Section 3]. We proceed similar as in the proof for [26, Lemma 5.2]. Fix t∈(0,T)such that u(·,t)=u0(·)on ∂ in the sense of traces. We wish to show that lim r→0− Br(γ )∩∩h uh(x,t)−u(x,t)dx≤ω(|h|)(5.2)
R. Rainer et al. for Hn−1-almost every γ∈∂( ∩h). To this end, let γ∈∂( ∩h)⊂(∂ ∩h)∪(∂h∩) ∪(∂ ∩∂h). First case: γ∈∂ ∩h.Letr>0 be so small that Br(γ ) ⊂h. Then − Br(γ )∩ uh(x,t)−u(x,t)dx=− Br(γ )∩ uh(x,t)−u0(γ ) dx +− Br(γ )∩ u0(γ ) −u(x,t)dx =− Br(γ )∩ u(x−h,t)−u0(γ ) dx +− Br(γ )∩ u0(γ ) −u(x,t)dx ≤− Br(γ )∩ ω(|x−h−γ|)dx +− Br(γ )∩ u0(γ ) −u(x,t)dx, where the last inequality follows from (5.1). Since ωis continuous, the first integral at the right-hand side converges to ω(|h|)as r→0. The second integral on the other hand converges to zero for Hn−1-almost every γ∈∂. Thus we obtain (5.2). Second case: γ∈∂h∩.Letr>0besosmallthatBr(γ ) ⊂. Then − Br(γ )∩h uh(x,t)−u(x,t)dx =− Br(γ −h)∩ u(x,t)−u(x+h,t)dx =− Br(γ −h)∩ u(x,t)−u0(γ −h)dx +− Br(γ −h)∩ u0(γ −h)−u(x+h,t)dx ≤− Br(γ −h)∩ u(x,t)−u0(γ −h)dx +− Br(γ −h)∩ ω(|x+h−(γ −h)|)dx, where the last inequality follows from (5.1) since γ−h∈∂.Asr→0, the first integral at the right-hand side converges to zero for Hn−1-almost every γ∈∂ and the second integral converges to ω(|h|). Thus we obtain (5.2). Third case: γ∈∂ ∩∂h∩∂( ∩h).By[26, Lemma 3.1] for Hn−1-almost every γ∈∂ ∩∂h∩∂( ∩h)it holds that tru(γ, t)=tr∩hu(γ, t)=u0(γ ), trhu(γ −h,t)=tr∩hu(γ −h,t)=u0(γ −h). (5.3)
An evolutionary Haar-Rado type theorem We have − Br(γ )∩∩h uh(x,t)−u(x,t)dx=− Br(γ )∩∩h u(x−h,t)−u0(γ −h)dx +− Br(γ )∩∩h u0(γ ) −u(x,t)dx +− Br(γ )∩∩h u0(γ −h)−u0(γ ) dx. As r→0, the first two integrals on the right-hand side converge to zero by (5.3). Moreover, we have u0(γ −h)−u0(γ ) ≤ω(|h|)by ω-continuity. Thus we obtain (5.2). Lemma 5.2. Suppose the conditions (2.1)-(2.3)are in force. Further suppose that u is a variational solution to the Cauchy-Dirichlet problem (1.1)on Tand that u0is bounded in . Then u is bounded in T. Proof. We only show that uis bounded from above as a lower bound follows analogously. Let M:= supx∈|u0(x)|. We show that for large enough β>0, the function a(x,t):= M+βt is a variational super-solution on T. Since a≥uon ∂PT, it follows from the comparison principle (Lemma 3.3) that u≤a≤M+βTa.e. in T. It now remains to show that ais a variational super-solution on T.Tothisaim, let v∈a+Lp([0,T]; W1,p 0(T)) ∩L2(T),v≥awith ∂tv∈L2(T)be a test function. By convexity of ffor any ξ∈Rnthere is λξ∈Rnsuch that f(ξ +ζ) ≥f(ξ) +λξ·ζ. We define ϕ(x,t):= v(x,t)−a(x,t)and use the above with ξ:= Da(x,t)=0 and ζ(x,t):= Dϕ(x,t)to obtain f(Dv) =f(Da +Dϕ) ≥f(Da)+λ·Dϕ. Integrating this over Twe get T f(Dv)dz≥T f(Da)+λ·Dϕdz=T f(Da)dz,(5.4) where the last identity follows from the Gauss-Green theorem since ϕ(·,t)∈ W1,p 0() for almost all t(cf. [13, Theorem 4.6]). Observe then that ∂tv(v −a)=1 2∂t(v −a)2+∂ta(v −a).
R. Rainer et al. Thus T ∂tv(v −a)+g(x,v)−g(x,a)dz−1 2T (v −a)2(x,·)dxT 0 =T 1 2∂t(v −a)2+∂ta(v −a)+g(x,v) −g(x,a)dz−1 2 (v −a)2(x,·)dxT 0 ≥T ∂ta(v −a)+(v −a)g+ u(a,x)dz,(5.5) where in the last inequality we also used the convexity of g(x,·). In the view of (5.4) and (5.5) it now suffices to show that ∂ta(v −a)+(v −a)g+ u(a,x)=(β +g+ u(M+βt,x))(v −a)≥0. Since v−a≥0 and by convexity g+ u(M+βt)≥g+ u(M), this follows by taking β≥|g+ u(M)|. We are now ready to finish the proof of our main theorem. Proof of Theorem 2.4.Fix h∈Rn. Then by Lemma 5.1 we have uh(·,t)−ω(|h|)≤u(·,t)on ∂( ∩h) in the sense of traces for almost all t∈(0,T). Moreover, by ω-continuity of u0we have uh(·,0)−ω(|h|)≤u(·,0)in ∩h. Since uis bounded in Tby Lemma 5.2, it follows from Theorem 4.2 that the functionuh−ω(|h|)isavariationalsub-solutionon(∩h)T.Thusthe comparison principle (Lemma 3.3) implies that uh−ω≤ua.e. in ( ∩h)T. Let (x0,t0), (y0,t0)∈Tbe Lebesgue points of u.Taker>0 so small that both Qr(x0,t0),Qr(y0,t0)⊂T.Leth=x0−y0. Then we have Qr(x0,t0)⊂ ( ∩h)T. Thus − Qr(x0,t0) uh−udz≤− Qr(x0,t0) ω(|h|)dz, that is, − Qr(x0,t0) ω(|x0−y0|)dz≥− Qr(x0,t0) u(x−x0+y0,t)−u(x,t)dz =− Qr(y0,t0) u(x,t)dz−− Qr(x0,t0) u(x,t)dz. Letting r→0, this implies that ω(|x0−y0|)≥u(y0,t0)−u(x0,t0).
An evolutionary Haar-Rado type theorem Proof of Corollary 2.5.For (x,t)∈Tand γ∈∂,wehave u(x,t)−u0(γ ) ≤l2(x)−u0(γ ) =l2(x)−l2(γ ) ≤ω(|x−γ|). Similarly, u(x,t)−u0(γ ) ≥l1(x)−u0(γ ) =l1(x)−l1(γ ) ≥−ω(|x−γ|). Thus |u(x,t)−u0(γ )|≤ω(|x−γ|)for a.e. (x,t)∈Tand γ∈∂. Thus we can apply Theorem 2.4, which shows the claim. 6. An application with the bounded slope condition In this section we apply the Haar-Rado type theorem to obtain the spatial Lipschitz continuity of variational solutions, provided that f∈C2(Rn)is -uniformly convex and u0satisfies the bounded slope condition. We begin with the following lemma, which says that affine functions can be bent so that they become suitable barriers. The definition of the barrier function is in the spirit of Bousquet and Brasco [8, Proposition 4.2]. Lemma 6.1. Let g be as in (2.2)and (Hh,ω). Suppose that is convex and that f∈C2(Rn)is uniformly -convex. Let x0∈∂ and let a :Rn→Rbe an affine function. Then there is β>0and η∈Rn,|η|=1, such that a(x):= a(x)+β(x−x0)·η(3diam−(x−x0)·η), is a variational super-solution in T. Moreover, a≥ain. Proof. Since is convex, we can rotate and translate it so that x0=0 and ⊂ {x∈Rn:x1>0}.Weset a(x):= a(x)+βx1(3diam−x1). Letv≥abeatest function forthevariational inequality. Toseethataisavariational super-solution, we need to show that T f(Dv) −f(Da)dz +T ∂tv(v −a)+g(x,v)−g(x,a)dz −1 2 (v −a)2(x,·)dxT 0≥0.(6.1) Since ∂ta=0, we have ∂tv(v −a)=1 2∂t(v −a)2,
R. Rainer et al. whence T ∂tv(v −a)+g(x,v)−g(x,a)dz−1 2 (v −a)2(x,·)dxT 0 =T g(x,v)−g(x,a)dz.(6.2) Furthermore, since a≤v, it follows from convexity of g(x,·)that g(x,v)−g(x,a)≥(v −a)g+ u(x,a)almost everywhere in T.(6.3) Thus in the view of (6.1), (6.2) and (6.3) it suffices to show that f(Dv) −f(Da)+(v −a)g+ u(x,a)dz≥0.(6.4) Since fis C2and convex, for any ξ∈Rnwe have f(ξ +ζ) ≥f(ξ) +ζ·Df(ξ) for all ζ∈Rn. We set ϕ(x,t):= v(x,t)−a(x,t)and apply the above with ξ:= Da(x)and ζ:= Dϕ(x,t). We obtain f(Da(x)+Dϕ(x,t)) =f(ξ +ζ) ≥f(ξ) +ζ·Df(ξ) =f(Da(x)) +Dϕ(x,t)·Df(Da(x)) for almost all (x,t)∈T. Integrating this over Twe get T f(Dv)dz=T f(Da+Dϕ)dz ≥T f(Da)+Dϕ·Df(Da)dz =T f(Da)−ϕdivxDf(Da)dz =T f(Da)−(v −a)divxDf(Da)dz,(6.5) where the identity follows from the Gauss-Green theorem. By (6.4) and (6.5)it now suffices to show that T (g+ u(x,a)−divxDf(Da))(v −a)dz≥0.(6.6) Recall that the condition (Hh,ω) says that c≥d+ω(|h|)⇒ g+ u(y−h,c)≥g+ u(y,d)for all h,y∈Rnand c,d∈R.
An evolutionary Haar-Rado type theorem Wefix apoint z0∈whose choicedoesnotmatter.Usingthe above withc:= a(x), d=a(x)−ω(|h|),h:= z0−xand y:= z0, we obtain g+ u(x,a(x)) ≥g+ u(z0,a(x)−ω(|h|)) ≥g+ u(z0,a(x)−ω(diam )) ≥g+ u(z0,−aL∞() −ω(diam )), where we also used that a≥ain . Since v−a≥0, condition (6.6) further reduces to showing that g+ u(z0,−aL∞() −ω(diam )) −divxDf(Da(x)) ≥0 (6.7) for almost all x∈. We observe that Da is a constant vector and compute Da(x)=D(a(x)+βx1(3diam−x1)) =Da +3β(diam )e1−2βx1e1 so that D2a(x)=−2β(e1⊗e1). Hence by chain rule and -uniform convexity of fwe have −divxDf(Da(x)) =−tr(JDf◦Da(x)) =−tr(JDf (Da(x))JDa(x)) =2βtr(D2f(Da(x))e1⊗e1) ≥2β(|Da(x)|). (6.8) Observe that |Da(x)|=|Da +3β(diam )e1−2βx1e1|≥3βdiam −2βx1−|Da| =β(3diam−2x1)−|Da|. Thus we can make |Da|arbitrarily large in by taking large enough β.Usingthe condition s(s)→∞as s→∞(with s=|Da|), it follows that there exists β=β(,,g,aW1,∞())such that 2β(|Da(x)|)=2β |Da(x)||Da(x)|(|Da(x)|) ≥2β |Da|+5βdiam |Da(x)|(|Da(x)|) ≥|g+ u(x0,−aL∞() −ω(diam ))|.(6.9) The desired inequality (6.7) now follows from (6.8) and (6.9). Proof of Corollary 2.9.By Corollary 2.5, it suffices to show that there exist Lipschitz continuous functions l1,l2∈W1,p u0() such that l1≤u≤l2in T.
R. Rainer et al. We will only construct the function l2as the construction of l1is analogical. Let x0∈∂. Since u0∂ satisfies the bounded slope condition with a constant Qand u0is Lipschitz in , there is an affine function ax0:Rn→Rsuch that u0(x0)=ax0(x0)and u0≤ax0in , and additionally |Dax0|≤max(Q,||Du0||L∞()),see[4, Lemma 2.3]. Let ax0be the variational super-solution given by Lemma 6.1. Then we have u0(x0)=a(x0)=ax0(x0)and u0≤ax0≤ax0in . Thus it follows from comparison principle (Lemma 3.3) that u≤ax0in T. Moreover, from the proof of Lemma 6.1 we see that ||Dax0||L∞() depends only on ,g,,sup ∂ |u0|and |Dax0|. In other words, the Lipschitz constant of ax0is independent of x0. Therefore the function l2(x):= inf x0∈∂ax0(x), x∈, is a desired Lipschitz continuous upper bound for u. Note that l2is in fact Lipschitz since it is the infimum of a family of Lipschitz functions with a uniform Lipschitz constant. 7. Declarations. Funding R. Rainer has been supported by the FWF-Project P 31956 “Doubly Nonlinear Evolution Equations”. Conflicts of interest/Competing interests Not applicable. Availability of data and material Not applicable. Code availability Not applicable.
An evolutionary Haar-Rado type theorem Authors’ contributions Each author contributed equally to the scientific value of this paper. All authors commented and took part in improvements on previous versions of the manuscript and approved the final version. Funding Open access funding provided by Paris Lodron University of Salzburg. 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/. References [1] Bögelein, V., Dacorogna, B., Duzaar, F., Marcellini, P., Scheven, C.: Integral convexity and parabolic systems. SIAM J. Math. Anal. 52(2), 1489–1525 (2020) [2] Bögelein, V., Duzaar, F., Marcellini, P.: Existence of evolutionary variational solutions via the calculus of variations. J. Differ. Equ. 256(12), 3912–3942 (2014) [3] Bögelein, V., Duzaar, F., Marcellini, P., Scheven, C.: Doubly nonlinear equations of porous medium type. Arch. Ration. Mech. Anal. 229(2), 503–545 (2018) [4] Bögelein, V., Duzaar, F., Marcellini, P., Signoriello, S.: Parabolic equations and the bounded slope condition. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(2), 355–379 (2017) [5] Bögelein, V., Stanin, T.: The one-sided bounded slope condition in evolution problems. Ann. Mat. Pura Appl. 4 199(2), 573–587 (2019) [6] Bousquet, P.: Boundary continuity of solutions to a basic problem in the calculus of variations. Adv. Calc. Var. 3, 1–27 (2010) [7] Bousquet, P.: Continuity of solutions of a problem in the calculus of variations. Calc. Var. 41, 413–433 (2011) [8] Bousquet, P., Brasco, L.: Global Lipschitz continuity for minima of degenerate problems. Math. Ann. 366(3–4), 1403–1450 (2016) [9] Bousquet, P., Clarke, F.: Local lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Equ. 243, 489–503 (2007) [10] Cellina, A.: On the bounded slope condition and the validity of the euler lagrange equation. SIAM J. Control Optim. 40, 1270–1279 (2001) [11] Cellina, A.: Comparison results and estimates on the gradient without strict convexity. SIAM J. Control Optim. 46, 738–749 (2007) [12] Clarke, F.: Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4, 511–530 (2005)
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