The one-sided bounded slope condition in evolution problems

We establish a local Lipschitz regularity result of solutions to the Cauchy–Dirichlet problem associated with evolutionary partial differential equations ∂tu-divDf(∇u)=0,inΩT,u=u0,on∂PΩT.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\,\mathrm{div}\,}}Df(\nabla u) = 0, &{} \quad \text{ in } \, \Omega _T,\\ u=u_0, &{} \quad \text{ on } \, \partial _{{\mathcal {P}}}\, \Omega _T. \end{array} \right. \end{aligned}$$\end{document}We do not impose any growth assumptions from above on the function f:Rn→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$\end{document} and only require it to be convex and coercive. The domain Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is required to be bounded and convex, and the time-independent boundary datum u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0$$\end{document} is supposed to be convex and Lipschitz continuous on Ω¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\Omega }}$$\end{document}. It can be seen as an evolutionary analogue to the one-sided bounded slope condition. Additionally, assuming Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} to be uniformly convex, we establish global continuity on ΩT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega _T}$$\end{document} of the solution.


Introduction
In this article, we are concerned with regularity properties of solutions to a class of Cauchy-Dirichlet problems of evolutionary partial differential equations of the form ∂ × (0, T ). We do not require our variational integrand f to satisfy any growth conditions from above but merely assume it to be convex and-to guarantee existence of solutions-also coercive (see 2.1). This approach follows a famous result in Hilbert-Haar theory for stationary problems, which-starting with a paper by Haar [14] and followed by papers from Hartman and Nirenberg [16], Stampacchia [23], Miranda [22], and Hartman and Stampacchia [17]can now even be found in textbooks (see [13,Chapter 1] as one possible reference). Under the bounded slope condition (see Definition 5.2), it ensures the existence of Lipschitz continuous minimizers. For time-dependent problems, a related result for linear growth functionals can be found in [15]. Moreover, in [4] the existence of Lipschitz continuous solutions to parabolic Cauchy-Dirichlet problems is proven under the assumption that the time-independent initial and lateral boundary datum satisfies the bounded slope condition. More precisely, Lipschitz continuity with respect to the parabolic metric is obtained. In recent years, there were different attempts to weaken the quite restrictive bounded slope condition. In [11], Clarke introduced the one-sided bounded slope condition and proved local Lipschitz continuity of minimizers to elliptic variational functionals. Later, a different generalization-named by Mariconda and Treu in [20] as Cellina bounded slope conditionwas introduced by Cellina in [10] as a result of establishing a new kind of comparison principle working without the strict convexity assumption on the variational integrand in the elliptic case. Mariconda and Treu then combined the two approaches in [21] to receive a one-sided version of Cellina's condition. Further results on the one-sided and full bounded slope condition can be found in [7][8][9]19] and the references therein.
Our paper has its origin in the effort to weaken the bounded slope condition in the evolutionary setting. As in [11], we impose the one-sided bounded slope condition in order to compensate the lack of growth. As strict convexity of the variational integrand is not necessary to get unique solutions in the parabolic setting, we stay with Clarke's approach in this article and transfer the main ideas of his proof into the context of parabolic problems. To adapt the lower bounded slope condition to the new setting, we require existence of affine functions lying below the boundary datum u 0 at each point x 0 ∈ in contrast to just requiring it on ∂ as in [11,Section 1] in the elliptic case, so that we can cope with both the initial and lateral boundary parts at once. This condition is encoded into our structural assumptions on u 0 (see 2.3)-following an alternative characterization of Clarke's lower bounded slope condition in [11, Proposition 1.1]-and will be explicitly stated in Lemma 4.1.
Starting from the notion of variational solutions (see Definition 2.1), we prove local Lipschitz continuity with respect to space and time. The advantage of working with variational solutions rather than weak solutions is that they allow employing methods from the calculus of variations. If the integrand is sufficiently regular such that the Euler-Lagrange equation makes sense, it can be shown that variational solutions are weak solutions and vice versa, cf. [3,Sect. 1.3]. In the proof of the local Lipschitz continuity, we use a dilation method. Quite remarkably, this method allows to prove local Lipschitz continuity with respect to space and time. The convexity of the initial datum is needed to ensure the existence of an affine function lying below the solution in any parabolic boundary point (see Lemma 4.2). On the other hand, in [4] spatial Lipschitz continuity is proven under the two-sided bounded slope condition by a translation with respect to the spatial variable only. To prove Lipschitz continuity in time by a translation in time under the two-sided bounded slope condition would require the initial datum to be an affine function (u 0 as well as −u 0 have to be convex). In this respect, one of the advantages of the one-sided bounded slope condition in the parabolic setting is that for convex initial data solutions are Lipschitz continuous also with respect to the time variable.
Moreover, we are also interested in global regularity of variational solutions. Global Lipschitz continuity, however, cannot be expected. There are counterexamples already in the stationary setting [11,Sect. 1]. With this respect, it is more natural to investigate global continuity of solutions as in the stationary setting in [7,Theorem 5] and [11,Theorem 2.2]. In fact, as a second main result we establish that variational solutions are continuous on T , provided that the domain is uniformly convex. In a certain sense, our results allow to weaken the assumptions in the main existence result in [4], which imposes the full bounded slope condition on the Cauchy-Dirichlet boundary datum and already starts in a suitably regular Lipschitz class of functions. As mentioned before, the consequence is that solutions are not anymore Lipschitz continuous up to the boundary, but merely locally Lipschitz and continuous up to the boundary in the uniformly convex case.

Notation and main results
Throughout this paper, T := × (0, T ) denotes a spacetime cylinder over a bounded and convex domain ⊂ R n and T > 0. For short, we often write v(t) := v(·, t) for a function in L 1 ( T ) ≡ L 1 (0, T ; L 1 ( )). We consider the parabolic Cauchy-Dirichlet problem (1.1) with an integrand f : R n → R that is only supposed to be convex and coercive. More precisely, we assume that with some ν > 0, p > 1 and μ ∈ R. Since we do not impose a growth assumption from above on f , the notion of weak solution might in general not be well defined. Therefore, we introduce the notion of variational solution which goes back to a paper by Lichnewsky and Temam [18].
We identify holds true, for any comparison function v ∈ u 0 + L p (0, T ; W If the integrand f satisfies additional assumptions like p-growth from above, then variational solutions are weak solutions in the usual sense, cf. [3]. Moreover, the variational inequality (2.2) ensures that the initial datum u(0) = u 0 (0) is assumed. In this paper, we are mainly interested in time-independent initial and lateral boundary data u 0 : → R. We assume that u 0 is convex, u 0 is Lipschitz continuous with constant L > 0. ( Note that the second condition of (2.3) together with the convexity of f ensures that f (∇u o ) dx < ∞ and u o ∈ L 2 ( ), which are the hypotheses in [3,Theorem 1.2]. Contrary to the elliptic setting, in the parabolic setting strict convexity is not a necessary assumption for uniqueness of solutions. In fact, as stated above, convexity is enough, cf. [4] and Lemma 3.4. Our first main result ensures that variational solutions are locally Lipschitz continuous with respect to space and time.
where δ := dist(C, ∂ T ) denotes the Euclidean distance of C to the boundary of T and The next natural question concerns global regularity of variational solutions. Global Lipschitz continuity cannot be expected due to the counterexample in [11, page 5] and [7]. Nevertheless, under the stronger uniform convexity condition on the domain we are able to prove continuity of variational solutions up to the lateral boundary. We first give the definition of uniform convexity from [22, Definizione 6.1].

Definition 2.4 A bounded, open subset
⊂ R n is called uniformly convex if for every boundary point x 0 ∈ ∂ there exists a hyperplane H x 0 passing through that point satisfying for any y ∈ ∂ . (2.4) Note that this assumption is stronger than strict convexity. As shown in [22, Proposizione 6.2], this condition ensures that any C 2 -function satisfies the bounded slope condition on ∂ . Our second main result can then be stated as follows: Theorem 2.5 Suppose that is uniformly convex, that the variational integrand f : R n → R satisfies (2.1), that the Cauchy-Dirichlet datum u 0 fulfills (2.3) and that u is a variational solution in the sense of Definition 2.1. Then, u is continuous on T .

Setup and preliminaries
Throughout the paper, we will use some important results for variational solutions that will be stated here for the convenience of the reader. We start with localization principles for variational solutions with respect to space and time. The proofs are straight forward and can for instance be deduced as in [4, Lemma 3.2 and Remark 4.2].

Lemma 3.1 (Spatial localization principle) Let u be a variational solution on T according to Definition 2.1 with ∂ t u ∈ L 2 ( T ) and let
⊂ be a subdomain. Then, u is also a variational solution on the subcylinder T := × (0, T ).

Lemma 3.2 (Temporal localization principle) Let u be a variational solution on T according to Definition
Next, we observe that the variational inequality (2.2) can be reformulated by the use of the time derivative of u.

Remark 3.3 Let u be a variational solution according to Definition
Adding and subtracting T ∂ t u(v−u)dxdt on the right-hand side of the variational inequality (2.2) and using integration by parts with respect to time, we find that By an approximation argument, we infer that the preceding inequality is valid for any Another very important result, which will be used frequently throughout the paper, is the following version of the comparison principle for variational solutions. Its proof can be deduced as in [4,Lemma 4.3].
Lemma 3.4 Suppose that f : R n → R is a variational integrand satisfying (2.1) and that for i ∈ {1, 2} are two variational solutions on T according to Definition 2.1 with u 1 ≤ u 2 on ∂ P T in the sense of traces. Then, we have u 1 ≤ u 2 a.e. on T .

Local Lipschitz continuity
In this section, we prove our first main result, the local Lipschitz continuity of variational solutions stated in Theorem 2.3. We start with a simple observation for convex functions. For the sake of completeness, we include the proof.
Proof We first consider the case x 0 ∈ . From the convexity of the function ϕ, we deduce that the subdifferential ∂ϕ(x 0 ) of ϕ in x 0 is nonempty. For any subgradient p ∈ ∂ϕ(x 0 ) ⊂ R n of ϕ at x 0 , we have Due to the Lipschitz continuity of ϕ with constant L > 0, we know that Then, there exists a (not re-labeled) subsequence and p ∈ R n such that p i → p. Since | p i | ≤ L for any i ∈ N we also have | p| ≤ L and due to the continuity of ϕ we conclude that ϕ(x) ≥ ϕ(x 0 ) + p, x − x 0 . This finishes the proof of the lemma.
Combining this result with the comparison principle in Lemma 3.4, we obtain the following lemma.
for every x ∈ , where the affine function x 0 is defined in the statement of the lemma. In particular, this shows that u ≥ x 0 on ∂ P T . Note that the constant in time extension of x 0 to T is a variational solution in the sense of Definition 2.1 with initial and lateral boundary values x 0 and that ∂ t u ∈ L 2 ( T ) due to Theorem 2.2. Therefore, the application of the comparison principle in Lemma 3.4 yields that u ≥ x 0 a.e. in T .
We are now in the position to prove Theorem 2.3. The idea to use a dilation of the domain in order to prove local Lipschitz continuity of minimizers to elliptic variational integrals is due to Clarke [11]. Here, we introduce a parabolic version of this technique. Quite surprisingly, we obtain a stronger result than local Lipschitz continuity with respect to the parabolic metric. We prove that variational solutions are locally Lipschitz continuous in space and time.
Step 1: Bound for the difference u λ −u on the parabolic boundary ∂ P λ T . Let λ ∈ (0, 1), ξ ∈ , and τ ∈ [0, T ]. We claim that in the sense of traces, where , and u λ := u ξ,τ ;λ . From Lemma 4.2, we infer that there exists p ∈ R n with | p| ≤ L such that u ≥ x 0 a.e. in T , where x 0 denotes the affine function This completes the proof of the claimed inequality (4.4). In the next step, we extend inequality (4.4) to the whole dilated spacetime cylinder λ T .
Step 2: Bound for the difference u λ −u on the spacetime cylinder λ T . Let λ ∈ (0, 1), ξ ∈ , and τ ∈ [0, T ]. We claim that where M 0 is defined in (4.5). As before, we use the shorthand notations u λ := u ξ,τ ;λ , and λ T := ξ,τ ;λ T and recall that ∂ t u ∈ L 2 ( T ) by Theorem 2.2. Due to the spatial and temporal localization principles for variational solutions from Lemmas 3.1 and 3.2, we know that (u + M 0 (1 − λ))| λ T is a variational solution on λ T ⊂ T . We claim that also u λ is a variational solution on λ T . To prove this, we consider a comparison function v ∈ u 0 + L p ( Keeping in mind that 0 λ = (1 − λ 2 )τ and T λ = λ 2 T + (1 − λ 2 )τ by (4.2), the computation shows the claim that u λ is a variational solution on λ T . Thus, so far we have shown that u + M 0 (1 − λ) and u λ are variational solutions on λ T with weak time derivatives in L 2 ( λ T ). Moreover, from (4.4) we know that in the sense of traces. At this point, the comparison principle from Lemma 3.4 yields the claimed inequality (4.6).
Step 3: Local spatial Lipschitz continuity. Here, we prove that For x = y, we denote by π ∂ (x|y) the spatial projection of x onto ∂ in the direction of y, which is the unique point of the form π ∂ (x|y) = x + α(y − x) ∈ ∂ for some α > 1. Then d ∂ (x|y) = |x − π ∂ (x|y)|. We now choose ξ = π ∂ (x|y) and τ = t in the dilations defined in (4.1) -(4.3). Then, y can be written in the form and t = t τ ;λ . From (4.6), we infer that holds true for a.e. x, y ∈ and a.e. t ∈ (0, T ). The comparison principle in Lemma 3.4 ensures that |u| ≤ u 0 L ∞ ( ) a.e. in T . Hence, adding (1 − λ)u(x, t) on both sides of the preceding inequality, we obtain for a.e. x, y ∈ and a.e. t ∈ [0, T ). Since the argument can be repeated with x and y reversed and choosing ξ = π ∂ (y|x), the claimed local Lipschitz estimate (4.7) follows.
Step 4: Lipschitz continuity with respect to time. In the final step, we prove that for a.e. x ∈ and a.e. s, t ∈ (0, T ) with s < t, where M is defined as in (4.8).
For the dilations defined in (4.1)-(4.3), we choose ξ = x and τ = 0, so that x = x ξ ;λ and From (4.6), we infer that holds true for a.e. x ∈ and a.e. 0 < s < t < T . As before, we add (1 − λ)u(x, t) on both sides of the preceding inequality and recall that |u| ≤ u 0 L ∞ ( ) in T . In this way, we obtain for a.e. x ∈ and a.e. 0 < s < t < T . Next, we keep ξ = x and choose τ = T in the dilations (4.1)-(4.3). Then, we have In an analogous way as done above, we infer that Combining these two estimates, we receive the claimed inequality (4.9). Note that (4.9) is equivalent with for a.e. x ∈ and a.e. s, t ∈ (0, T ). This finishes the proof of Theorem 2.3.

Continuity up to the boundary
In this section, we analyze global continuity of variational solutions. We start with a result ensuring lower Lipschitz semi-continuity up to the parabolic boundary.
In particular, u is lower Lipschitz semi-continuous when approaching the parabolic boundary of T .
Proof Let (x 0 , t 0 ) ∈ ∂ P T . From Lemma 4.2, we know that there exists p ∈ R n with | p| ≤ L such that u ≥ x 0 a.e. in T , where x 0 denotes the affine function defined by This implies the claim, since Concerning global Lipschitz continuity, Lemma 5.1 is the best result we can expect. Already in the elliptic setting the counterexample in [11,Sect. 1] shows that global Lipschitz continuity is in general not true, i.e., variational solutions might fail to be upper Lipschitz semi-continuous when approaching the parabolic boundary. Therefore, we have to restrict ourselves to the proof of global continuity as stated in Theorem 2.5. This will be achieved in the remaining part of this section.
As mentioned before, due to a result by Miranda [22] we know that the uniform convexity of ensures that every C 2 -function satisfies the bounded slope condition on ∂ . For the sake of completeness, we shortly recall the classical bounded slope condition.

Definition 5.2
We say that a function ϕ : ∂ → R satisfies the bounded slope condition (B.S.C.) with constant Q > 0 if for any x 0 ∈ ∂ there exist two affine functions − x 0 and + From [22, Proposizione 6.2], we have the following result.

Proposition 5.3
Let be a uniformly convex domain in R n . Then, every function ϕ ∈ C 2 (R n ) satisfies the B.S.C. on ∂ .
We now come to the proof of our second main result, stated in Theorem 2.5. We first prove the result under the stronger condition that u 0 satisfies the full bounded slope condition.

Proposition 5.4
Suppose that f : R n → R is a variational integrand satisfying (2.1) and let u 0 ∈ W 1,∞ ( ) be a Cauchy-Dirichlet datum such that u 0 | ∂ fulfills the B.S.C. with constant Q > 0. Then, there exists a variational solution u : T → R in the sense of Definition 2.1 which in addition is Lipschitz continuous with respect to the parabolic metric on T , i.e., u ∈ C 0;1,1/2 ( T ).
Proof Following the notation in [4, Chapter 1.1], we define and denote by K u 0 ( T ) the subclass of those v ∈ K ( T ) coinciding with u 0 on the lateral boundary ∂ × (0, T ). From here on, we will continue in two steps, first proving existence of a variational solution under C 1 -regularity of f and afterward dealing with general convex integrands via an approximation procedure.
Step 1: Integrands of class C 1 . We assume that f satisfies (2.1) and is continuously differentiable on R n . An application of [4, Theorem 1.2] yields the existence of a unique variational solution u ∈ K u 0 ( T ) in the sense of [4, Definition 1.1] satisfying and the variational inequality Our next aim is to show that u is a variational solution in the sense of Definition 2.1 with initial and lateral boundary values u 0 . Since To this aim, we derive the weak Euler-Lagrange equation associated with (5.2) by choosing the comparison function v = u + sϕ with s ∈ (0, 1) and ϕ ∈ C ∞ ( T ) such that ϕ = 0 on ∂ × (0, T ); we note that ∂ t u ∈ L 2 ( T ) due to [4,Theorem 1.3]. In this way, we obtain Due to convexity of f and the fact that f is of class C 1 , we have Using this estimate in the previous inequality and letting s ↓ 0, which is allowed since f (∇u + s∇ϕ) is uniformly bounded on T , we receive for all ϕ ∈ C ∞ ( T ) with ϕ = 0 on ∂ × (0, T ). Finally, replacing ϕ by −ϕ, we also get the reverse inequality. Hence, (5.3) holds as an identity. By approximation, we observe that this identity holds for a larger class of test functions. More precisely, we have Using again the convexity of f in the form yields the inequality In the second term on the right-hand side, we write u = v −(v −u) and perform an integration by parts with respect to time. Simplifying the appearing expressions leaves us with . This shows that u is a variational solution in the sense of Definition 2.1. Finally, applying [4,Theorem 1.3] in the case of a C 1 -integrand f , we have u being Lipschitz continuous with respect to the parabolic metric. More precisely, as in [4,Sect. 8] we have for any parabolic cylinder where in the last line we used (5.1). The proof of this inequality can be deduced as in [6,Lemma 3.1]. Observe that the argument continues to work for cylinders Q (z 0 ) intersecting the initial boundary ×{0}, cf. [5,Lemma 4.13]. We note that is convex by assumption, and therefore, it is a Lipschitz domain. This allows to apply Poincaré's inequality for parabolic cylinders with center z 0 = (x 0 , t 0 ) on the lateral boundary, i.e., with x 0 ∈ ∂ . In this way, we obtain for any parabolic cylinder Q (z 0 ) with x 0 ∈ ∂ and t 0 ∈ [0, T ). Combining both cases, we find that holds true with a constant c = c(n, ) and for any parabolic cylinder Q (z 0 ) with z 0 ∈ T . Due to the parabolic version of Campanato's characterization of Hölder continuity by Da Prato [12, Teorema 3.1], this implies the Lipschitz continuity of u with respect to the parabolic metric, i.e., u ∈ C 0;1,1/2 ( T ).
Step 2: General convex integrands. Now we consider the case where f satisfies only assumption (2.1). From the convexity assumption, we infer that f is Lipschitz continuous on B 2M (0), where M is defined in (5.1). We let K := sup B 2M (0) |D f | < ∞. For ε ∈ (0, 1), we infer from [1, Corollary 1.3] the existence of a convex function f ε : R n → R of class C 1 with f −ε ≤ f ε ≤ f and sup B M (0) |D f ε | ≤ K . Note that the coercivity of f ε follows immediately from the one of f and the fact that f ε ≥ f − ε. Applying our previously obtained result from Step 1 with integrand f ε , we infer that there exist variational solutions in the sense of Definition 2.1 with integrand f ε instead of f satisfying (5.1) and (5.4), i.e., hold true for any parabolic cylinder Q (z 0 ) with z 0 ∈ T . Recall that the preceding inequality is a bound on the parabolic Campanato seminorm of u ε , which ensures-using Da Prato's characterization [12, Teorema 3.1] again-that u ε is Lipschitz continuous with respect to the parabolic metric. Moreover, we have the quantitative estimate with a constant C independent of ε. Moreover, from the comparison principle in Lemma 3.4 we know that Our next aim is to pass to the limit ε ↓ 0. Inequalities (5.6) and (5.7) ensure that the family of functions {u ε } ε>0 is uniformly bounded and equicontinuous with respect to the Euclidean metric on R n+1 . Therefore, Arzelà-Ascoli's theorem ensures the existence of a subsequence (u ε i ) i∈N with ε i ↓ 0 as i → ∞ and a continuous function u ∈ C 0 ( T ) such that u ε i → u uniformly on T as i → ∞. Since we can now pass to the limit ε i ↓ 0 in inequality (5.6), we infer that u ∈ C 0;1,1/2 ( T ). Particularly, we have that u ∈ C 0 ([0, T ]; L 2 ( )) ∩ u 0 + L p (0, T ; W 1, p 0 ( )). Our last step is to show that u fulfills the variational inequality (2.2) associated with the original variational integrand f . To this aim, we make use of the lower semi-continuity of u → T f (∇u) dxdt with respect to strong convergence in L 1 ( T ) (cf. [2, Lemma 3.1]), the variational inequality for u ε , f − ε ≤ f ε ≤ f , as well as the uniform convergence of u ε i to u. For v ∈ u 0 + L p (0, T ; W 1, p 0 ( )) with v(0) ∈ L 2 ( ) and ∂ t v ∈ L 2 ( T ), we receive which finishes the proof of Proposition 5.4.
Observe that under the full B.S.C. we have obtained global Lipschitz continuity with respect to the parabolic metric in Proposition 5.4. As explained before, this cannot be expected under the assumption of the one-sided bounded slope condition.
Proof of Theorem 2. 5 Here, we assume that is uniformly convex. Since u 0 is Lipschitz continuous on , there exists a sequence of C 2 -functions (u (i) 0 ) i∈N such that u Due to the uniform convergence u (i) 0 → u 0 , we have that ε i → 0 as i → ∞. Using the comparison principle from Lemma 3.4, we find that |u − u (i) | ≤ ε i a.e. on T . Therefore, we know u to be a uniform limit of a sequence of continuous functions on T ; hence, it is continuous itself on T . This proves the claim of Theorem 2.5.