The obstacle problem for degenerate doubly nonlinear equations of porous medium type

We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation ∂tb(u)-div(Df(Du))=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(Df(Du)) = 0, \end{aligned}$$\end{document}where the nonlinearity b:R≥0→R≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}$$\end{document} is increasing, piecewise C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} and satisfies a polynomial growth condition. The prototype is b(u):=um\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(u) := u^m$$\end{document} with m∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in (0,1)$$\end{document}. Further, f:Rn→R≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :\mathbb {R}^n \rightarrow \mathbb {R}_{\ge 0}$$\end{document} is convex and fulfills a standard p-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.


Introduction and results
Let Ω ⊂ ℝ n be a bounded Lipschitz domain and (0, T) with 0 < T < ∞ a finite time interval. In the following, Ω T ∶= Ω × (0, T) denotes a space-time cylinder. The prototype of the equations considered in the present paper is with parameters m ∈ (0, ∞) and p ∈ (1, ∞) . For m = 1 and p ∈ (1, ∞) , the preceding equation reduces to the parabolic p-Laplace equation, while it is known as the porous medium equation if m ∈ (0, ∞) and p = 2 . Based on the behavior of solutions, doubly nonlinear equations can be subdivided into slow diffusion equations with p − 1 > m and fast diffusion equations with p − 1 < m . Further, we distinguish between doubly degenerate equations ( p > 2 , 0 < m < 1 ), singular-degenerate equations ( 1 < p < 2 , 0 < m < 1 ), 1 3 degenerate-singular equations ( p > 2 , m > 1 ) and doubly singular equations ( 1 < p < 2 , m > 1 ), cf. [18]. The porous medium equation and related doubly nonlinear equations are relevant in models for fluid dynamics, filtration and soil science, cf. [4-6, 21, 28]. In the present paper, we are concerned with the obstacle problem to doubly nonlinear equations of doubly degenerate and singular-degenerate type. In order to treat (1.1), we use an approach that originates from Lichnewsky and Temam [23] and has later been developed by Bögelein, Duzaar, Marcellini and Scheven [8][9][10][11] to cover a wide range of parabolic problems. More precisely, we are concerned with variational solutions to the Cauchy-Dirichlet problem associated with (1.1) for given initial and boundary values g ∶ Ω T → ℝ ≥0 , i.e., functions u ∶ Ω T → ℝ ≥0 satisfying the variational inequality associated with (1.2) for any admissible comparison map v ∶ Ω T → ℝ ≥0 . Here, we used the abbreviation for u, v ∶ Ω → ℝ ≥0 . Formally, the variational inequality can be derived by multiplying (1.2) by v − u , where v ∶ Ω T → ℝ ≥0 coincides with u on the lateral boundary Ω × (0, T) and then integrating the result over Ω T . For the diffusion part, we use integration by parts and the convexity of 1 p | ⋅ | p . Finally, by integration by parts the time derivative is shifted from u to v, leading in particular to the integrals over the top and bottom of the space-time cylinder on the right-hand side of the variational inequality. In the present paper, we impose an additional pointwise obstacle condition of the form u ≥ for some obstacle function ∶ Ω T → ℝ ≥0 . This means that u is a variational solution to the obstacle problem associated with Eq. (1.2) and initial and boundary values g ∶ Ω → ℝ ≥0 if u coincides with g on the parabolic boundary (Ω × {0}) ∪ ( Ω × (0, T)) and satisfies the obstacle condition u ≥ a.e. in Ω T and the preceding variational inequality holds true for any comparison map v with boundary values g and v ≥ a.e. in Ω T . For the precise definition, cf. Definition 1.1. At this stage, some words on the history of the problem are in order. First, the seminal work of Grange and Mignot [17] and Alt and Luckhaus [3] should be mentioned. In [3], the authors were among other things concerned with the obstacle problem associated with doubly nonlinear equations of the type where b is the gradient of a convex C 1 -function with b(0) = 0 , (b(z), ) is continuous in z and and fulfills an ellipticity and (p − 1)-growth condition with respect to the gradient variable and (b(z)) is continuous in z and satisfies a suitable growth condition. Further, a two-sided obstacle condition is imposed with obstacle functions ± ∈ L p (0, T;W 1,p (Ω)) ∩ L ∞ (Ω T ) with t ± ∈ L 1 (Ω T ) and − ≤ + a.e. in Ω T . Under these assumptions, the existence of variational solutions has been established via time discretization for Neumann boundary values. However, the proof extends to the case of an additional Dirichlet boundary condition with zero boundary values on a part of the boundary. Later, Bernis [7] showed the existence of weak solutions to the Cauchy problem associated with higher order doubly nonlinear equations on unbounded domains. Further, Ivanov, Mkrtychyan and Jäger [18][19][20] used regularization and a priori Hölder estimates to prove the existence of regular weak solutions to the Cauchy-Dirichlet problem associated with doubly nonlinear equations. The boundary values satisfy g ∈ W 1,p (Ω T ) ∩ L ∞ (Ω T ) and an additional continuity assumption with respect to space and time. A different approach has been pursued by Akagi and Stefanelli [2] in order to treat the Cauchy-Dirichlet problem with homogenous Dirichlet boundary values associated with doubly nonlinear equations of the type where b ⊂ ℝ × ℝ and ⊂ ℝ n × ℝ n are maximal monotone graphs that fulfill polynomial growth conditions. The authors solve the problem by means of elliptic regularization (Weighted Energy Dissipation Functional method) after transforming it into the dual formulation − div ( (Db −1 (v)) ∋ − t v . Recently, by a nonlinear version of the method of minimizing movements Bögelein, Duzaar, Marcellini and Scheven [11] were able to prove the existence of nonnegative variational solutions to the Cauchy-Dirichlet problem with time-independent boundary values associated with where b ∶ ℝ ≥0 → ℝ ≥0 is continuous, piecewise C 1 and satisfies a polynomial growth condi- x ∈ Ω and f satisfies a coercivity, but not necessarily a growth condition. This allows f to have nonstandard growth like exponential or (p, q)-growth with 1 < p < q < ∞ . Note that the required nonnegativity of the solutions is an obstacle condition with obstacle function ≡ 0 . In the case of singly nonlinear equations of p-Laplace type, Bögelein, Duzaar and Scheven [12] established the existence of variational solutions to the obstacle problem with a far more general obstacle function ∈ L 2 (Ω T ) ∩ L p (0, T;W 1,p (Ω)) and time-dependent boundary values via the classical method of minimizing movements. Finally, the author has been concerned with the singular equation/system where m > 1 , f is convex and satisfies a standard p-growth and coercivity condition. By the nonlinear minimizing movements scheme developed in [11] and suitable approximation arguments, the existence of signed or vector-valued variational solutions to the Cauchy-Dirichlet problem with time-dependent boundary values and the existence of solutions to the obstacle problem with time-dependent obstacle function have been established, cf. [26,27]. More precisely, the boundary values and the obstacle function are contained in the space L p (0, T;W 1,p (Ω)) with time derivative in L 1 (0, T;L m+1 (Ω)) and initial values in L m+1 (Ω) . In the present paper, the question of uniqueness will not be discussed, since this is a delicate and widely open issue for doubly nonlinear equations. We refer to [15] for a counterexample and to [3,15] for sufficient conditions.

The general doubly nonlinear equation
In the present paper, we are concerned with the doubly nonlinear equation Here, we assume that f ∶ ℝ n → ℝ ≥0 is a Borel-measurable, convex function that fulfills the growth and coercivity condition with constants 0 < ≤ L for all ∈ ℝ n . Observe that (1.3) and the convexity of f together imply that f is locally Lipschitz continuous. More precisely, holds true for any , ∈ ℝ n , cf. [24,Eq. (2.9)]. Further, the nonlinearity b ∶ ℝ ≥0 → ℝ ≥0 is continuous and piecewise C 1 in ℝ >0 . Replacing b(u) by b(u) − b(0) , we suppose without loss of generality that b(0) = 0 . Moreover, we assume that there exist constants 0 < ≤ m ≤ 1 such that holds true whenever u > 0 , b(u) > 0 and b � (u) exists. In particular, this implies that b � (u) ≥ 0 if it exists. Then, the primitive of b defined by is a convex C 1 function with Φ(0) = 0 . Further, the convex conjugate (Fenchel conjugate) of Φ is defined by which immediately implies Fenchel's inequality Since Φ is convex, we easily compute that equality holds for v = b(u) , i.e.
At this stage, we define for any u, v ≥ 0 . In the variational inequality associated with (1.2), we will use boundary terms for functions u, v ∶ Ω → ℝ ≥0 . Furthermore, in order to be able to formulate solutions to the obstacle problem, we define the Orlicz space related to Φ and some domain A ⊂ ℝ k , k ∈ ℕ , by For details on Orlicz spaces, we refer to the monographs [1,25]. By the assumptions on b, we obtain that both Φ and Φ * satisfy the ∇ 2 and the Δ 2 condition (see (2.3)). In particular, the Δ 2 condition on Φ implies that an equivalent definition of the Orlicz space above is given by Henceforth, we often abbreviate the modular (see [25,Chapter III.3.4]) by In the present paper, we assume that L Φ (A) is equipped with the Orlicz norm which is equivalent to the Luxemburg norm Dealing with these norms is not always straightforward. However, since Φ fulfills the Δ 2 condition, norm convergence is equivalent to modular convergence, i.e., holds true for any v i , v ∈ L Φ (A) , i ∈ ℕ , and for sets S ⊂ L Φ (A) we know that cf. [25,Chapter III.3.4]. Analogously, we define the Orlicz space L Φ * (A) , the modular * A (⋅) and the norms ‖ ⋅ ‖ L Φ * (A) and ‖ ⋅ ‖ � L Φ * (A) related to the convex conjugate Φ * . In this setting, the generalized Hölder's inequality [25,Chapter III.3.3]. Further, the Δ 2 condition on Φ implies that L Φ (Ω) is separable and that the dual space of (L Φ

The main result
In order to formulate a boundary condition, we consider the affine parabolic space g + L p (0, T;W 1,p 0 (Ω)) consisting of the functions v ∈ L p (0, T;W 1,p (Ω)) such that for a.e. t ∈ (0, T) . In the present paper, we assume that nonnegative boundary values g ∶ Ω T → ℝ ≥0 are given by and that the nonnegative obstacle function ∶ Ω T → ℝ ≥0 satisfies At this stage, we are able to state the main result of the present paper. Note that we can conclude from (1.11) that u attains the initial datum g o in the L Φ -sense; see Lemma 2.18.

Methods of proof
First, in Sect. 2, we collect lemmas that we need in the subsequent proofs of the existence theorems. Their proofs are already known or easy. Next, in Sect. 3, we prove a preliminary existence result for regular data, i.e., boundary values and an obstacle with time derivative in L 2 (Ω T ) ∩ L p (0, T;W 1,p (Ω)) and initial values in L 2 (Ω) ∩ W 1,p (Ω) . Since b � (0) is infinite, we assume that g and are bounded away from zero. The proof relies on a nonlinear version of the method of minimizing movements. More precisely, we fix a step size h K ∶= T∕K for some K ∈ ℕ and consider time slices of Ω T at the time points ih K , i ∈ {0, … , K} . Then, we set u 0 = g(0) and iteratively define minimizers u i of the elliptic variational functionals (1.9) g ∈ L p (0, T;W 1,p (Ω)) with t g ∈ L 1 (0, T;L Φ (Ω)) and g o ∶= g(0) ∈ L Φ (Ω) if b(u) = u and hence the scheme reduces to the classical method of minimizing movements in the linear case. Next, in Sect. 3.2, we derive suitable energy estimates for the minimizers u i . Here, the stronger assumptions on the data are crucial. As in the classical scheme, we assemble the functions u i to a map u (K) ∶ Ω × (−h K , T] → ℝ ≥0 that is piecewise constant with respect to time by setting u (K) (t) ∶= u i for t ∈ ((i − 1)h K , ih K ] , i ∈ {0, … , K} . By the energy estimates from Sect. 3.2 and the compactness result 2.21, we find a subsequence and a suitable limit map u ∈ L ∞ (0, T;L Φ (Ω)) ∩ g + L p (0, T;W 1,p 0 (Ω)) such that u (K) ⇁ u weakly in L p (0, T;W 1,p (Ω)) and u (K) → u a.e. in Ω T . In Sect. 3.4, we assemble the functionals F i such that u (K) inherits a minimizing property and thus deduce a preliminary variational inequality for u (K) . Finally, in Sect. 3.5, we pass to the limit K → ∞ in these preliminary inequalities, which allows us to show that u is the desired variational solution. In Sect. 4, we relax the regularity assumptions on the spatial variables of the data. More precisely, the time derivatives of the boundary values and obstacle are now contained in L 2 (0, T;L Φ (Ω)) ∩ L p (0, T;W 1,p (Ω)) and the initial values in L Φ (Ω) . Further, g and may attain the value zero. The proof of the existence result relies on standard mollification of the boundary values and obstacle with respect to the spatial variables. Since the regularized data g and satisfy the assumptions of Sect. 3, we find variational solutions u , > 0 , corresponding to g and . By the energy bound from Lemma 2.20, we deduce that a subsequence converges weakly to a suitable limit map u ∈ L ∞ (0, T;L Φ (Ω)) ∩ g + L p (0, T;W 1,p 0 (Ω)) . Passing to the limit ↓ 0 in the variational inequalities fulfilled by u , we conclude that u is the desired variational inequality to g and . To this end, it is important to understand that weak convergence u * ⇁ u weakly in L ∞ (0, T;L Φ (Ω)) as ↓ 0 in general does not imply b(u ) * ⇁ b(u) weakly in L ∞ (0, T;L Φ * (Ω)) as ↓ 0 . Even if there is a convergent subsequence, the limit might not be b(u). Therefore, we need to use a technique similar to the one in [13,Lemma 9.1] to establish the desired convergence assertion. Finally, in Sect. 5, we give the proof of Theorem 1.2. The technique is similar to the one in Sect. 4, but based on the time mollification procedure described in Sect. 2.3 instead of standard mollification.

Technical lemmas
In this section, we collect some lemmas that we will need for the proof of the existence result. For the proofs of the lemmas 2.1, 2.3, 2.6, 2.7, 2.9 and 2.10, we refer to [11, Section 2.1].

Lemma 2.1 For any continuous, piecewise
and any > 1 , u > 0 , we have that:

Lemma 2.5 For any
In particular, we have that

Lemma 2.11 Assume that the functions
in Ω with respect to the n-dimensional Lebesgue measure L n .

Difference quotients
First, adapting the proof of [16, Theorem 1.33], we show the following variant of Lebesgue's differentiation theorem.

Lemma 2.12
Let (X, ‖ ⋅ ‖ X ) be a separable Banach space and v ∈ L 1 (0, T;X) . Then, for By Lebesgue's differentiation theorem, we conclude that Combining the preceding considerations, we infer Since > 0 was arbitrary, this yields the claim. ◻ Let h > 0 . The difference quotient of a function v with respect to time is denoted by We prove the following convergence assertion for X = L Φ (Ω).

Lemma 2.13
Then, we have that Observe that Therefore, choosing suitable i ∈ {1, … , k} and applying Lemma 2.12 with X = L Φ (Ω) we compute that a.e. in [0, T] as k → ∞ . Moreover, in a similar way, we find that

Lemma 2.14
Then, the following integration by parts formula holds true, where the error terms 1 (h) and 2 (h) are given by Proof For the proof of the integration by parts formula, we refer to [11,Lemma 2.10]. It remains to show the second assertion of the lemma. By Lemma 2.7, we conclude that ) . Therefore, we find that Next, by the generalized Hölder's inequality (1.8) and Hölder's inequality, we compute that as h ↓ 0 . This concludes the proof of the lemma. ◻

Mollification in time
In addition to standard mollification, we also consider the following mollification technique introduced by Landes [22]. We construct the regularization [v] h , h > 0 , to a given function v, such that it formally solves the ordinary differential equation The precise construction is as follows. Let X be a separable Banach space and v o ∈ X ; in the applications, we will have X = L r (Ω) with r ≥ 1 and

Lemma 2.15 Suppose that X is a separable Banach space and
In the case r = ∞ , the bracket […] 1 r in the preceding inequality has to be interpreted as 1. Moreover, in the case we have the following assertion.

The initial condition
Here, we show that variational solutions attain the initial datum g o in the L Φ -sense. For the proof of the following statement, we refer to [ and (1.7), we have that Furthermore, we know that t g ∈ L 1 (0, T;L Φ (Ω)) . Altogether, discarding the nonnegative energy term on the left-hand side, taking the square root, integrating over ∈ (0, h) for h ∈ (0, T) and dividing the result by h we conclude that in the limit h ↓ 0 . Next, from (2.3) and the convexity of Φ , we deduce that holds true for a.e. t ∈ [0, T] . Since g ∈ C 0 ([0, T];L Φ (Ω)) , we have that g(t) → g o strongly in L Φ (Ω) as t ↓ 0 and hence by (1.6) that Φ(|g(t) − g o |) → 0 as t ↓ 0 . Thus, we conclude that Hence, using the estimate from Lemma 2.9, Hölder's inequality and Lemma 2.7, we infer By u, g ∈ L ∞ (0, T;L Φ (Ω)) and (1.7), the first term on the right-hand side of the preceding inequality is bounded. Therefore, combining the preceding inequality with (2.8), we find that Altogether, combining (2.9), (2.10) and (2.11), we obtain that in the limit h ↓ 0 . This concludes the proof of the lemma. ◻

An energy bound
In this section, we derive an energy bound for variational solutions.
Proof If u is a variational solution and v an admissible comparison map in the sense of (2.14) At this stage, we choose ∶= 6m . This allows us to reabsorb the term from the right-hand side of the preceding inequality into the left-hand side, which yields the claim. ◻

Compactness
The proof of following result can be found in [11,  hold true for all k ∈ ℕ , and that u (k) ⇁ u weakly in L p (0, T;W 1,p (Ω)) as k → ∞ . Then, there exists a subsequence ⊂ ℕ such that in the limit ∋ k → ∞ we have the following convergences:

A sequence of minimizers to elliptic variational functionals
Fix a step size h ∈ (0, 1] such that h = T∕K for some K ∈ ℕ . For i ∈ {0, … , K} define g i ∶= g(ih) and i ∶= (ih) . Set u 0 = g o . Then, we inductively find minimizers u i of the functionals in the class v ∈ L Φ (Ω) ∩ g i + W 1,p 0 (Ω) with v ≥ i . Note that this class is not empty, since v = g i is admissible. The existence of minimizers u i is ensured, for example, by the direct method of the calculus of variations. For convenience of the reader, we give the precise proof. Proof Consider a minimizing sequence (u j ) j∈ℕ , i.e. By means of Lemmas 2.3 and 2.6, we find that

3
Together with (1.3) and the fact that h ∈ (0, 1] , this implies Hence, by (1.7) the sequence (u j ) j∈ℕ in bounded in L Φ (Ω) ∩ W 1,p (Ω) . Thus, there exists a (not relabeled) subsequence and a limit map u ∈ L Φ (Ω) ∩ g * + W 1,p 0 (Ω) such that in the limit j → ∞ . Observe that the obstacle condition u ≥ * a.e. in Ω is preserved. Since F is convex and lower semicontinuous with respect to strong convergence in L Φ (Ω) ∩ W 1,p (Ω) by means of Fatou's Lemma, F is also lower semicontinuous with respect to weak convergence in L Φ (Ω) ∩ W 1,p (Ω) , cf. [14, Corollary 3.9]. As a consequence, we find that which yields the claim. ◻

Energy estimates
Observe that v ∶= u i−1 + i − i−1 ≥ i is an admissible comparison function for u i . Using the minimality of u i with respect to F i , we obtain that where the definition of I and II is clear in this context. First, we estimate I . To this end, using the Lipschitz estimate (1.4), Young's inequality, the assumption h ≤ 1 and the coercivity assumption (1.3), we conclude that where we used the short-hand notation for any ∈ (0, T] . Reabsorbing into the left-hand side of the preceding inequality, we have that In order to estimate the right-hand side of the preceding inequality, we iterate (3.3) from j = 1, … , i for i = 1, … , k − 1 . This yields Inserting this into (3.4), we obtain that . .

Convergence to a limit map
In the following, we set h K ∶= T∕K for K ∈ ℕ . We define the piecewise constant function Analogously, we define g (K) and (K) . Combining estimate (3.5) with the coercivity condition (1.3) 1 and discarding the nonnegative sum on the left-hand side, we find that Hence, the sequence (u (K) ) K∈ℕ is bounded in L ∞ (0, T;W 1,p (Ω)) . Since u (K) ∈ g (K) + L ∞ (0, T;W 1,p 0 (Ω)) , there exists a subsequence ⊂ ℕ and a limit map u ∈ g + L p (0, T;W 1,p 0 (Ω)) such that in the limit ∋ K → ∞ . By Lemma 2.7 and the energy estimates (3.5) and (3.6), the assumptions of Lemma 2.21 are satisfied for the sequence (u (K) ) K∈ . Therefore, choosing another subsequence still denoted by , we obtain that in the limit ∋ K → ∞ . At this stage, observe that (K) → a.e. in Ω T as k → ∞ . Combining this with (3.8) 2 and the fact that u (K) ≥ (K) a.e. in Ω T , we deduce that u satisfies the obstacle condition u ≥ a.e. in Ω T . Next, by means of Lemma 2.7 and (3.5), we conclude that Thus, we find a subsequence such that Δ h K √ Φ u (K) ⇁ w weakly in L 2 (Ω T ) . In order to characterize w, we use this fact together with (3.8) 1 . More precisely, we obtain for any ∈ C ∞ 0 (Ω T ) that By a density argument, this ensures that w = t √ Φ(u) and in particular t . This implies that Φ(u) ∈ C 0 ([0, T];L 1 (Ω)) and hence by means of Lemma 2.9 that u ∈ C 0 ([0, T];L Φ (Ω)).

Minimizing property of the approximations
Observe that for each K ∈ ℕ , the map u (K) is a minimizer of the functional in the class of functions v ∈ L Φ (Ω T ) ∩ g (K) + L p (0, T;W 1,p 0 (Ω)) satisfying v ≥ (K) a.e. in Ω T . Indeed, using the definitions of (K) and u (K) and the minimality of u i with respect to F i , we compute for any admissible function v as above Note that for any fixed comparison map v ∈ L Φ (Ω T ) ∩ g (K) + L p (0, T;W 1,p 0 (Ω)) with v ≥ (K) a.e. in Ω T and any s ∈ (0, 1) the convex combination w s ∶= u (K) + s(v − u (K) ) of u (K) and v is still admissible, since (K) ≤ w s ∈ L Φ (Ω T ) ∩ g (K) + L p (0, T;W 1,p 0 (Ω)) . Then, the minimality of u (K) and the convexity of f imply that for any s ∈ (0, 1) , with equality for s = 0 . Reabsorbing ∬ Ω T (1 − s)f Du (K) dxdt into the left-hand side of the preceding inequality and dividing by s, we find that holds true. Note that the map s ↦ 1 s Φ(w s ) − Φ u (K) is monotone and converges a.e. in Ω T to the L 1 (Ω T )-function b u (K) v − u (K) , since Φ is convex. Passing to the limit s ↓ 0 in the preceding inequality with the aid of the dominated convergence theorem, we deduce that in Ω T . Note that in particular u (K) (t) = g o for t ∈ (−h K , 0] . Using the comparison map (0,T) v + [ ,T] u (K) for some ∈ (0, T] instead of v, we infer the localized variational inequality for any ∈ (0, T] and v ∈ L Φ (Ω ) ∩ g (K) + L p (0, ;W 1,p 0 (Ω)) with v ≥ (K) a.e. in Ω .

Variational inequality for the limit map
Here, we pass to the limit K → ∞ in (3.9) in order to deduce the variational inequality for the limit map. To this end, we consider an arbitrary map v ∈ g + L p (0, T;W is an admissible comparison map in (3.9), since v k ≥ (K) and v K ∈ L Φ (Ω T ) ∩ g (K) + L p (0, T;W 1,p 0 (Ω)) are satisfied. Hence, we obtain that Now, we consider the terms of (3.10) separately. First, by (3.7) and since f is convex and satisfies the coercivity condition (1.3) 1 , we have that By (1.4) and the fact that D (K) → D in L p (Ω T , ℝ n ) , we conclude that in the limit K → ∞ . Next, shifting the difference quotient in the last term on the right-hand side from b u (K) to (K) − , we obtain that where the definition of I K − III K is clear in this context. By the generalized Hölder's inequality (1.8) and Hölder's inequality, we find that Combining the energy estimate (3.6) with Lemma 2.3 and recalling (1.7), we conclude that b u (K) (t) K∈ℕ is bounded in L ∞ (0, T;L Φ * (Ω)) . Further, by Lemma 2.13 and since Next, by the generalized Hölder's inequality (1.8), Lemma 2.12, the definition of (K) and since ∈ C 0 ([0, T];L Φ (Ω)) , we deduce that (3.14) lim Similarly, we obtain that In order to treat the remaining term in (3.10), we apply the finite integration by parts formula from Lemma 2.14. This yields where we used the abbreviations Furthermore, the error terms 1 (h K ) and 2 (h K ) are given by For the characterization of B 0 (h K ) and 2 (h K ) , we used that u (K) (t) = g o and v(t) = v(0) for t ∈ (−h K , 0] . Since t v ∈ L 1 (0, T;L Φ (Ω)) , Lemma 2.14 implies that Next, we consider the first term on the right-hand side of (3.17). Since we have that ) and by (3.8) 2 , we know that b u (K) * ⇁ b(u) weakly in L ∞ (0, T;L Φ * (Ω)) . Therefore, Since we are not allowed to pass to the limit ∋ K → ∞ in B (h K ) directly, we integrate (3.10) over ∈ (t o , t o + ) for some ∈ (0, T) and t o ∈ [0, T − ] and divide the result by . Combining this with (3.17), we find that

Existence for less regular data with respect to the spatial variables
Next, we prove an existence result for boundary values and obstacle, whose time derivative is less regular with respect to the spatial variables. More precisely, we assume that 0 ≤ ≤ g a.e. on Ω T , and For the proof of the desired result, we need the following lemma, cf. [12,Lemma 4.3].
Since spt ( ) = Ω , we conclude that ∈ g + L p (0, T;W 1,p 0 (Ω)) and (0) ∈ L 2 (Ω) ∩ g o, + W 1,p 0 (Ω) . Further, since and are independent of time, we have that and Finally, by a standard property of the applied mollification procedure, we find that 0 < ≤ ≤ g . More generally, for any comparison map v ∈ g + L p (0, T;W and v ≥ a.e. in Ω T , we define the mollification Then, we obtain that v ∈ g + L p (0, T;W and v ≥ a.e. in Ω T . Next, we prove that Indeed, we conclude that in the limit ↓ 0 . Further, (4.3) in particular implies that there exists a (not relabeled) subsequence such that v → v a.e in Ω T . A similar computation shows that Moreover, observe that Hence, by Remark 2.4 the sequence (b(v )) >0 is bounded in L ∞ (0, T;L Φ * (Ω)) . Together, the preceding considerations prove that there exists another subsequence such that Next, we compute that in the limit ↓ 0 , which yields and a.e in Ω as ↓ 0.

3
Finally, we show that To this end, we first compute The second term on the right-hand side of the preceding inequality clearly vanishes as ↓ 0 . For the first term, by definition of and Lemma 4.1 we conclude that in the limit ↓ 0 . This proves (4.8). where the constant C is defined by By (4.5) and (1.7), (4.7) and (4.8) together with the growth condition (1.3) C is finite. Therefore, there exists a (not relabeled) subsequence and a limit map u ∈ L ∞ (0, T;L Φ (Ω)) ∩ g + L p (0, T;W 1,p 0 (Ω)) such that as ↓ 0 . By (4.3) applied to v = and (4.10) 2 , we obtain that

Improved convergence of the solutions
Next, we need to establish Since (b(u )) is bounded in L ∞ (0, T;L Φ * (Ω)) by (4.10) and Remark 2.4, there exists a subsequence such that for some limit map w ∈ L ∞ (0, T;L Φ * (Ω)) . Therefore, it remains to prove that w has the structure b(u). To this end, for h > 0 we consider mollifications [u − ] h and [u − ] h according to (2.7) with zero initial values and define Since L Φ (Ω) is separable, by Lemma 2.15 we obtain that w ,h , w h ∈ L ∞ (0, T;L Φ (Ω)) . Further, we find that w ,h ∈ g + L p (0, T;W 1,p 0 (Ω)) and w h ∈ g + L p (0, T;W 1,p 0 (Ω)) . Since u ≥ , we have that w ,h ≥ . Next, note that (2.6) implies Thus, by (4.10) 2 and since → in L p (0, T;W 1,p (Ω)) , we deduce that for any fixed h > 0 the sequence ( t [u − ] h ) is bounded in L p (Ω T ) . Further, by Lemma 2.17 holds true for fixed h > 0 . Therefore, from Rellich's theorem, we infer We did not have to pass to a subsequence, since the limit is determined by (4.13). Next, we use w ,h as comparison map in the variational inequality satisfied by u . Discarding the nonnegative terms ∬ Ω T f (Du ) dxdt and [u (T), w ,h (T)] , we deduce that which is equivalent to → Φ( (0) u → u in L min{ +1,p} (Ω T ) as ↓ 0.
according to (2.7) with v o = v(0) . In particular, we know that i.e., the mollification of t v according to (2.7) with zero initial datum. Therefore, by Lemma 2.15, we obtain that tṽi → t v in L 1 (0, T;L Φ (Ω)) as i → ∞ , which allows us to compute Finally, we establish the assertionṽ a.e. and in L Φ (Ω) as i → ∞.    By (5.2) together with (1.6), (5.4) and (5.5) together with the growth condition (1.3), C is finite. Hence, there exists a limit map u ∈ L ∞ (0, T;L Φ (Ω)) ∩ g + L p (0, T;W 1,p 0 (Ω)) and a (not relabeled) subsequence such that in the limit i → ∞ . The obstacle condition is preserved as i → ∞ . Indeed, by (5.8) and (5.2) applied to v i = i , we find that

Convergence of solutions
In this step, we wish to establish Since (b(u i )) i∈ℕ is bounded in L ∞ (0, T;L Φ * (Ω)) by Lemma 2.3, (1.7) and (5.7), we know that there exists w ∈ L ∞ (0, T;L Φ * (Ω)) such that (for a subsequence) However, it remains to prove that w has the structure b(u). To this end, let > 0 and consider mollifications [u i − i ] and [u − ] according to (2.7) with zero initial datum. Then, we define By Lemma 2.15 we obtain that w i, ∈ L ∞ (0, T;L Φ (Ω)) ∩ g i + L p (0, T;W 1,p (Ω)) and w ∈ L ∞ (0, T;L Φ (Ω)) ∩ g + L p (0, T;W 1,p (Ω)) . Furthermore, (2.6) implies that Since i → in L p (Ω T ) , by (5.8) 2 and Lemma 2.17, the sequence ([u i − i ] ) i∈ℕ is bounded in L p (Ω T ) for any fixed > 0 . Further, by (5.5), (5.8) 2 and Lemma 2.17 holds true for fixed > 0 . Therefore, we conclude from Rellich's theorem that for any fixed > 0 . Here, we did not have to pass to a subsequence, since the limit is determined by (5.11). Since u i ≥ i for any i ∈ ℕ , we know that w i, ≥ i . Using w i, as comparison map in the variational inequality for u i leads us to (5.8) u i * ⇁ u weakly * in L ∞ (0, T;L Φ (Ω)), u i ⇁ u weakly in L p (0, T;W 1,p (Ω))    The preceding inequality is equivalent to where the definition of I , II and III is clear in this context. By the generalized Hölder's inequality (1.8), (2.5) and Lemma 2.3, we estimate By (5.4) applied to v i = i , the first factor on the right-hand side of the preceding inequality stays bounded in the limit i → ∞ . Further, by the energy bound (5.7) we know that sup t∈[0,T] ∫ Ω Φ(u i ) dx ≤ 2C for any i ∈ ℕ . By Lemma 2.15, (5.2) and (5.8) 1 we find that Consequently, by (1.7), we conclude that sup t∈[0,T] ∫ Ω Φ(w i, ) dx is bounded by a constant independent of i ∈ ℕ and > 0 . Altogether, we obtain that with a constant c independent of i and . Next, by the growth condition (1.3), Lemma 2.15 and (5.5), we deduce that (5.13)  holds true for any admissible comparison map v ∈ g + L p (0, T;W 1,p 0 (Ω)) with t v ∈ L 1 (0, T;L Φ (Ω)) , v(0) ∈ L Φ (Ω) and v ≥ a.e. in Ω T . Passing to the limit ↓ 0 , we conclude that u is a variational solution to (1.2) in the sense of Definition 1.1. ◻