Abstract
We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation
where the nonlinearity \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is increasing, piecewise \(C^1\) and satisfies a polynomial growth condition. The prototype is \(b(u) := u^m\) with \(m \in (0,1)\). Further, \(f :\mathbb {R}^n \rightarrow \mathbb {R}_{\ge 0}\) is convex and fulfills a standard p-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
1 Introduction and results
Let \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain and (0, T) with \(0<T<\infty\) a finite time interval. In the following, \(\Omega _T := \Omega \times (0,T)\) denotes a space-time cylinder. The prototype of the equations considered in the present paper is
with parameters \(m \in (0,\infty )\) and \(p \in (1,\infty )\). For \(m=1\) and \(p \in (1,\infty )\), the preceding equation reduces to the parabolic p-Laplace equation, while it is known as the porous medium equation if \(m \in (0,\infty )\) 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 (\(1<p<2\), \(0<m<1\)), 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. [4,5,6, 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 \(g :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\), i.e., functions \(u :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) satisfying the variational inequality
associated with (1.2) for any admissible comparison map \(v :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\). Here, we used the abbreviation
for \(u,v :\Omega \rightarrow \mathbb {R}_{\ge 0}\). Formally, the variational inequality can be derived by multiplying (1.2) by \(v-u\), where \(v :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) coincides with u on the lateral boundary \(\partial \Omega \times (0,T)\) and then integrating the result over \(\Omega _T\). For the diffusion part, we use integration by parts and the convexity of \(\frac{1}{p}|\cdot |^p\). 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 \(u \ge \psi\) for some obstacle function \(\psi :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\). This means that u is a variational solution to the obstacle problem associated with Eq. (1.2) and initial and boundary values \(g :\Omega \rightarrow \mathbb {R}_{\ge 0}\) if u coincides with g on the parabolic boundary \((\Omega \times \{0\}) \cup (\partial \Omega \times (0,T))\) and satisfies the obstacle condition \(u \ge \psi\) a.e. in \(\Omega _T\) and the preceding variational inequality holds true for any comparison map v with boundary values g and \(v \ge \psi\) a.e. in \(\Omega _T\). 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 \(b(0)=0\), \({\mathbf {a}}(b(z),\xi )\) is continuous in z and \(\xi\) and fulfills an ellipticity and \((p-1)\)-growth condition with respect to the gradient variable and \({\mathbf {f}}(b(z))\) is continuous in z and satisfies a suitable growth condition. Further, a two-sided obstacle condition is imposed with obstacle functions \(\psi _\pm \in L^p(0,T;W^{1,p}(\Omega )) \cap L^\infty (\Omega _T)\) with \(\partial _t \psi _\pm \in L^1(\Omega _T)\) and \(\psi _- \le \psi _+\) a.e. in \(\Omega _T\). 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 \(g \in W^{1,p}(\Omega _T) \cap L^\infty (\Omega _T)\) 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 \(b \subset \mathbb {R}\times \mathbb {R}\) and \({\mathbf {a}} \subset \mathbb {R}^n \times \mathbb {R}^n\) 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 \(-{{\,\mathrm{div}\,}}({\mathbf {a}}(Db^{-1}(v)) \ni {\mathbf {f}} - \partial _t v\). 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 \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is continuous, piecewise \(C^1\) and satisfies a polynomial growth condition with \(b(0)=0\). Further, \((u,\xi ) \mapsto f(x,u,\xi )\) is convex for a.e. \(x \in \Omega\) 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 \(1<p<q<\infty\). Note that the required nonnegativity of the solutions is an obstacle condition with obstacle function \(\psi \equiv 0\). 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 \(\psi \in L^2(\Omega _T) \cap L^p(0,T;W^{1,p}(\Omega ))\) 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 \(L^p(0,T;W^{1,p}(\Omega ))\) with time derivative in \(L^1(0,T;L^{m+1}(\Omega ))\) and initial values in \(L^{m+1}(\Omega )\). 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.
1.1 The general doubly nonlinear equation
In the present paper, we are concerned with the doubly nonlinear equation
Here, we assume that \(f :\mathbb {R}^n \rightarrow \mathbb {R}_{\ge 0}\) is a Borel-measurable, convex function that fulfills the growth and coercivity condition
with constants \(0<\nu \le L\) for all \(\xi \in \mathbb {R}^n\). Observe that (1.3) and the convexity of f together imply that f is locally Lipschitz continuous. More precisely,
holds true for any \(\xi ,\eta \in \mathbb {R}^n\), cf. [24, Eq. (2.9)]. Further, the nonlinearity \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is continuous and piecewise \(C^1\) in \(\mathbb {R}_{>0}\). 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 \(0<\ell \le m \le 1\) such that
holds true whenever \(u>0\), \(b(u)>0\) and \(b'(u)\) exists. In particular, this implies that \(b'(u) \ge 0\) if it exists. Then, the primitive of b defined by
is a convex \(C^1\) function with \(\Phi (0)=0\). Further, the convex conjugate (Fenchel conjugate) of \(\Phi\) is defined by
which immediately implies Fenchel’s inequality
Since \(\Phi\) is convex, we easily compute that equality holds for \(v=b(u)\), i.e.
At this stage, we define
for any \(u,v \ge 0\). In the variational inequality associated with (1.2), we will use boundary terms
for functions \(u,v :\Omega \rightarrow \mathbb {R}_{\ge 0}\). Furthermore, in order to be able to formulate solutions to the obstacle problem, we define the Orlicz space related to \(\Phi\) and some domain \(A \subset \mathbb {R}^k\), \(k \in \mathbb {N}\), by
For details on Orlicz spaces, we refer to the monographs [1, 25]. By the assumptions on b, we obtain that both \(\Phi\) and \(\Phi ^*\) satisfy the \(\nabla _2\) and the \(\Delta _2\) condition (see (2.3)). In particular, the \(\Delta _2\) condition on \(\Phi\) 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 \(L^\Phi (A)\) is equipped with the Orlicz norm
which is equivalent to the Luxemburg norm
Dealing with these norms is not always straightforward. However, since \(\Phi\) fulfills the \(\Delta _2\) condition, norm convergence is equivalent to modular convergence, i.e.,
holds true for any \(v_i, v \in L^\Phi (A)\), \(i \in \mathbb {N}\), and for sets \(S \subset L^\Phi (A)\) we know that
cf. [25, Chapter III.3.4]. Analogously, we define the Orlicz space \(L^{\Phi ^*}(A)\), the modular \(\varrho _A^*(\cdot )\) and the norms \(\Vert \cdot \Vert _{L^{\Phi ^*}(A)}\) and \(\Vert \cdot \Vert '_{L^{\Phi ^*}(A)}\) related to the convex conjugate \(\Phi ^*\). In this setting, the generalized Hölder’s inequality
holds true for functions \(v \in L^\Phi (A)\), \(w \in L^{\Phi ^*}(A)\), cf. [25, Chapter III.3.3]. Further, the \(\Delta _2\) condition on \(\Phi\) implies that \(L^\Phi (\Omega )\) is separable and that the dual space of \((L^\Phi (A),\Vert \cdot \Vert _{L^\Phi (A)})\) is isometrically isomorphic to \((L^{\Phi ^*}(A),\Vert \cdot \Vert '_{L^{\Phi ^*}(A)})\), cf. [25, Chapters III, IV].
1.2 The main result
In order to formulate a boundary condition, we consider the affine parabolic space \(g + L^p(0,T;W^{1,p}_0(\Omega ))\) consisting of the functions \(v \in L^p(0,T;W^{1,p}(\Omega ))\) such that \(v(t) \in g(t) + W^{1,p}_0(\Omega )\) for a.e. \(t \in (0,T)\). In the present paper, we assume that nonnegative boundary values \(g :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) are given by
and that the nonnegative obstacle function \(\psi :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) satisfies
Definition 1.1
(Variational solution) Assume the convex integrand f satisfies (1.3) and that (1.9) and (1.10) hold true. A measurable nonnegative map \(u :\Omega _T \rightarrow \mathbb {R}_{\ge 0}\) in the class
is called a variational solution to the obstacle problem associated with (1.2) if and only if it solves the variational inequality
for a.e. \(\tau \in [0,T]\) and any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\).
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 \(L^\Phi\)-sense; see Lemma 2.18.
Theorem 1.2
Assume that the convex integrand f fulfills (1.3) and that the hypotheses (1.9) and (1.10) are satisfied. Then, there exists a variational solution
to (1.2) in the sense of Definition1.1. Furthermore, u attains the initial datum \(g_o\) in the \(L^\Phi\)-sense.
1.3 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 \(L^2(\Omega _T) \cap L^p(0,T;W^{1,p}(\Omega ))\) and initial values in \(L^2(\Omega ) \cap W^{1,p}(\Omega )\). Since \(b'(0)\) is infinite, we assume that g and \(\psi\) 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 \(K \in \mathbb {N}\) and consider time slices of \(\Omega _T\) at the time points \(ih_K\), \(i \in \{0,\ldots ,K\}\). Then, we set \(u_0 = g(0)\) and iteratively define minimizers \(u_i\) of the elliptic variational functionals
in the class \(v \in L^\Phi (\Omega ) \cap (g(ih_K) + W^{1,p}_0(\Omega ))\). Observe that \({\mathfrak {b}}[u,v] = \tfrac{1}{2} \Vert u-v\Vert _{L^2(\Omega )}^2\) 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 \(u^{(K)} :\Omega \times (-h_K,T] \rightarrow \mathbb {R}_{\ge 0}\) that is piecewise constant with respect to time by setting \(u^{(K)}(t) := u_i\) for \(t \in ((i-1)h_K,ih_K]\), \(i \in \{0,\ldots ,K\}\). By the energy estimates from Sect. 3.2 and the compactness result 2.21, we find a subsequence and a suitable limit map \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) such that \(u^{(K)} \rightharpoondown u\) weakly in \(L^p(0,T;W^{1,p}(\Omega ))\) and \(u^{(K)} \rightarrow u\) a.e. in \(\Omega _T\). 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 \(K \rightarrow \infty\) 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 \(L^2(0,T;L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega ))\) and the initial values in \(L^\Phi (\Omega )\). Further, g and \(\psi\) 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 \(g_\varepsilon\) and \(\psi _\varepsilon\) satisfy the assumptions of Sect. 3, we find variational solutions \(u_\varepsilon\), \(\varepsilon >0\), corresponding to \(g_\varepsilon\) and \(\psi _\varepsilon\). By the energy bound from Lemma 2.20, we deduce that a subsequence converges weakly to a suitable limit map \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega ))\big )\). Passing to the limit \(\varepsilon \downarrow 0\) in the variational inequalities fulfilled by \(u_\varepsilon\), we conclude that u is the desired variational inequality to g and \(\psi\). To this end, it is important to understand that weak convergence \(u_\varepsilon \mathop {\rightharpoondown }\limits ^{*}u\) weakly\(^*\) in \(L^\infty (0,T;L^\Phi (\Omega ))\) as \(\varepsilon \downarrow 0\) in general does not imply \(b(u_\varepsilon ) \mathop {\rightharpoondown }\limits ^{*}b(u)\) weakly\(^*\) in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) as \(\varepsilon \downarrow 0\). 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.
2 Preliminaries
2.1 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 \(C^1\) function \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) satisfying (1.5) and any \(\lambda >1\), \(u>0\) , we have that:
Remark 2.2
Assuming (1.5), we infer from Lemma 2.1 that
for any \(u>0\).
Lemma 2.3
Assume that b satisfies (1.5). Then,
holds true for any \(u \ge 0\).
Remark 2.4
Combining Lemma 2.3 with (1.7), we find that \(v \in L^\infty (0,T;L^\Phi (\Omega ))\) implies \(b(v) \in L^\infty (0,T;L^{\Phi ^*}(\Omega ))\).
Lemma 2.5
For any \(v \in L^\Phi (\Omega )\) , we have that
Proof
Set
If \(M \le 1\), by definition of the Luxemburg norm, we have \(\Vert v\Vert '_{L^{\Phi }(\Omega )} \le 1\). On the other hand, if \(M>1\), by Lemma 2.1, we find that
and therefore that \(\Vert v\Vert '_{L^{\Phi }(\Omega )} \le M^\frac{1}{\ell +1}\). Combining the cases implies (2.5). \(\square\)
Lemma 2.6
Assume that b satisfies (1.5). Then, for all \(u, v \ge 0\), we have that
Lemma 2.7
Assume that (1.5) is satisfied. Then, there exists a constant \(c=c(m,\ell )\) such that the estimates
hold true for all \(u,v \ge 0\).
Lemma 2.8
Assume that (1.5) is in force. If \((v_i)_{i \in \mathbb {N}}\) is a sequence in \(L^\Phi (\Omega )\) and \(v \in L^\Phi (\Omega )\) such that \(v_i \rightarrow v\) in \(L^\Phi (\Omega )\) as \(i \rightarrow \infty\), we also have that \(v_i \rightarrow v\) in \(L^{\ell +1}(\Omega )\).
Proof
The Lebesgue space \(L^{\ell +1}(\Omega )\) is obviously related to the function \(\Psi (x) := |x|^{\ell +1}\). By (2.3) for any \(\varepsilon >0\), we obtain that
Hence, \(\Phi\) is completely stronger than \(\Psi\), cf. [25, Definition 2.2.1]. By (1.6), we find that \(\lim _{i \rightarrow \infty } \varrho _\Omega (v_i-v) = 0\). Therefore, from the definition of the Luxemburg norm and [25, Theorem 5.3.1], we infer
This concludes the proof of the lemma. \(\square\)
Lemma 2.9
Assume that (1.5) holds true. If \((u_i)_{i \in \mathbb {N}}\) is a sequence in \(L^\Phi (\Omega )\) such that \(\Phi (u_i) \rightarrow \Phi (u)\) in \(L^1(\Omega )\) for some \(u \in L^\Phi (\Omega )\), , then \(u_i \rightarrow u\) in \(L^\Phi (\Omega )\) as \(i \rightarrow \infty\). In particular, we have that
Lemma 2.10
Suppose that b satisfies (1.5). If \(v \in L^1(\Omega _T)\) is given with \(\partial _t v \in L^\Phi (\Omega _T)\) and \(v(0) \in L^\Phi (\Omega )\), , we have that \(v \in C^0([0,T];L^\Phi (\Omega ))\).
Lemma 2.11
Assume that the functions \(v,\psi \in C^0([0,T];L^\Phi (\Omega ))\) satisfy \(v \ge \psi\) a.e. in \(\Omega _T\) with respect to the \((n+1)\)-dimensional Lebesgue measure \(\mathcal {L}^{n+1}\). Then, \(v(0) \ge \psi (0)\) a.e. in \(\Omega\) with respect to the n-dimensional Lebesgue measure \(\mathcal {L}^n\).
2.2 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 \((X,\Vert \cdot \Vert _X)\) be a separable Banach space and \(v \in L^1(0,T;X)\). Then, for a.e. \(t \in [0,T]\) , we have that
Proof
Since X is separable, there exists a dense subset \((v_i)_{i \in \mathbb {N}} \subset X\). Then, we know that \(t \mapsto \Vert v(t)-v_i\Vert _X \in L^1(0,T)\) for any \(i \in \mathbb {N}\). By Lebesgue’s differentiation theorem, we conclude that
holds true for any \(t \in [0,T] \setminus \mathcal {N}_i\), where \(\mathcal {N}_i\), \(i \in \mathbb {N}\), denotes a \(\mathcal {L}^1\)-null set. Consequently, \(\mathcal {N} := \bigcup _{i \in \mathbb {N}} \mathcal {N}_i \cup \{t \in [0,T] :v(t) \notin X\}\) is a \(\mathcal {L}^1\)-null set and
holds true for any \(t \in [0,T] \setminus \mathcal {N}\) and any \(i \in \mathbb {N}\). Next, fix \(t \in [0,T] \setminus \mathcal {N}\) and let \(\varepsilon >0\). Since \((v_i)_{i \in \mathbb {N}}\) is dense in X, there exists \(i \in \mathbb {N}\) such that
Combining the preceding considerations, we infer
Since \(\varepsilon >0\) was arbitrary, this yields the claim. \(\square\)
Let \(h>0\). The difference quotient of a function v with respect to time is denoted by
We prove the following convergence assertion for \(X=L^\Phi (\Omega )\).
Lemma 2.13
Assume that \(v \in C^0([0,T];L^\Phi (\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\). Further, let \(h_k :=T/k\) for some \(k \in \mathbb {N}\) and define the piecewise constant function \(v^{(k)} :\Omega _T \rightarrow \mathbb {R}\) by
Then, we have that
Proof
Fix \(t \in [0,T]\) and let \(i \in \{1,\ldots ,k\}\) such that \(t \in ((i-1)h_k,ih_k]\). Observe that
Therefore, choosing suitable \(i \in \{1,\ldots ,k\}\) and applying Lemma 2.12 with \(X=L^\Phi (\Omega )\) we compute that
a.e. in [0, T] as \(k \rightarrow \infty\). Moreover, in a similar way, we find that
in \(L^1(0,T)\) in the limit \(k \rightarrow \infty\). Here, we used that \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t}^{t+2h_k} \Vert \partial _t v(s)\Vert _{L^\Phi (\Omega )} \,\mathrm {d}s\) is the Steklov average of the function \(t \mapsto \Vert \partial _t v(t)\Vert _{L^\Phi (\Omega )} \in L^1(0,T)\). Hence, a version of the dominated convergence theorem (cf. [16, Theorem 1.20]) implies the claim. \(\square\)
The following statement is a slightly different version of the discrete integration by parts formula [11, Lemma 2.10].
Lemma 2.14
Let \(h \in (0,1]\), and \(u,v \in L^\Phi (\Omega \times (-h,T+h))\) be two nonnegative functions. Then, the following integration by parts formula
holds true, where the error terms \(\varvec{\delta }_1(h)\) and \(\varvec{\delta }_2(h)\) are given by
If we assume additionally that \(v \in L^\infty (-h_o,T+h_o;L^\Phi (\Omega ))\) and \(\partial _t v \in L^1(-h_o,T+h_o;L^\Phi (\Omega ))\) for some \(h_o>0\), then we have
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
First, observe that \(\big (b(v(t+h))-b(v(t))\big ) \Delta _h v(t) \rightarrow 0\) a.e. in \(\Omega _T\) as \(h \downarrow 0\). Further, we have that
By Remark 2.4, this implies that the sequence \(\big (b(v(t+h))-b(v(t))\big )_{h \in (0,h_o)}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Therefore, we find that
Combining this with \(\Delta _h v \rightarrow \partial _t v\) in \(L^1(0,T;L^\Phi (\Omega ))\) as \(h \downarrow 0\), we infer
Next, by the generalized Hölder’s inequality (1.8) and Hölder’s inequality, we compute that
as \(h \downarrow 0\). This concludes the proof of the lemma. \(\square\)
2.3 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
with initial condition \([v]_h(0) = v_o\). The precise construction is as follows. Let X be a separable Banach space and \(v_o \in X\); in the applications, we will have \(X=L^r(\Omega )\) with \(r\ge 1\) and \(X=L^\Phi (\Omega )\). Now, we consider \(v \in L^r(0,T;X)\) for some \(1 \le r \le \infty\), and define for \(h \in (0,T]\) and \(t \in [0,T]\) the mollification in time by
It is easy to check that \([v]_h\) satisfies (2.6). The basic properties of the mollifications \([\cdot ]_h\) are provided in the following Lemma, cf. [10, Appendix B] for the proofs of the statements.
Lemma 2.15
Suppose that X is a separable Banach space and \(v_o \in X\). If \(v \in L^r(0,T;X)\) for some \(r \ge 1\), then the mollification \([v]_h\) defined in (2.7) fulfills \([v]_h \in L^r(0,T;X)\) and for any \(t_o \in (0,T]\) there holds
In the case \(r=\infty\) , the bracket \([\ldots ]^\frac{1}{r}\) in the preceding inequality has to be interpreted as 1. Moreover, in the case \(r<\infty\) we have \([v]_h \rightarrow v\) in \(L^r(0,T;X)\) as \(h \downarrow 0\). Finally, if \(v \in C^0([0,T];X)\) and \(v_o=v(0)\), then \([v]_h \in C^0([0,T];X)\), \([v]_h(0) =v_o\), and moreover \([v]_h \rightarrow v\) in \(C^0([0,T];X)\) as \(h \downarrow 0\). \(\square\)
For maps \(v \in L^r(0,T;X)\) with \(\partial _t v \in L^r(0,T;X)\) we have the following assertion.
Lemma 2.16
Let X be a separable Banach space and \(r \ge 1\). Assume that \(v \in L^r(0,T;X)\) with \(\partial _t v \in L^r(0,T;X)\). Then, for the mollification in time defined by
, the time derivative can be computed by
and, moreover we have that
holds true. \(\square\)
Lemma 2.17
Let \(r \ge 1\), \((v_i)_{i \in \mathbb {N}}\) be a sequence in \(L^r(\Omega _T)\), \(v \in L^r(\Omega _T)\) and \(v_o \in L^r(\Omega )\). If \(v_i \rightharpoondown v\) weakly in \(L^r(\Omega _T)\) as \(i \rightarrow \infty\), then \([v_i]_h \rightharpoondown [v]_h\) weakly in \(L^r(\Omega _T)\) as \(i \rightarrow \infty\) holds true for the mollifications defined by (2.7) with fixed \(h>0\) and initial values \(v_o\).
2.4 The initial condition
Here, we show that variational solutions attain the initial datum \(g_o\) in the \(L^\Phi\)-sense. For the proof of the following statement, we refer to [11, Lemma 2.9].
Lemma 2.18
Any variational solution to (1.2) in the sense of Definition 1.1fulfills the initial condition \(u(0)=g_o\) in the \(L^\Phi\)-sense, i.e.
Proof
Since \(v=g\) is admissible in the variational inequality (1.11), by the generalized Hölder’s inequality (1.8), Lemmas 2.3 and 2.5, we find that
holds true for a.e. \(\tau \in [0,T]\). Recalling \(u,g \in L^\infty (0,T;L^\Phi (\Omega ))\) and (1.7), we have that
Furthermore, we know that \(\partial _t g \in L^1(0,T;L^\Phi (\Omega ))\). Altogether, discarding the nonnegative energy term on the left-hand side, taking the square root, integrating over \(\tau \in (0,h)\) for \(h \in (0,T)\) and dividing the result by h we conclude that
in the limit \(h \downarrow 0\). Next, from (2.3) and the convexity of \(\Phi\), we deduce that
holds true for a.e. \(t \in [0,T]\). Since \(g \in C^0([0,T];L^\Phi (\Omega ))\), we have that \(g(t) \rightarrow g_o\) strongly in \(L^\Phi (\Omega )\) as \(t \downarrow 0\) and hence by (1.6) that \(\Phi (|g(t)-g_o|) \rightarrow 0\) as \(t \downarrow 0\). Thus, we conclude that
Hence, using the estimate from Lemma 2.9, Hölder’s inequality and Lemma 2.7, we infer
By \(u,g \in L^\infty (0,T;L^\Phi (\Omega ))\) 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 \(h \downarrow 0\). This concludes the proof of the lemma. \(\square\)
Remark 2.19
In the case \(\Phi (v) = |v|^{m+1}\), i.e., \(L^\Phi (\Omega ) = L^{m+1}(\Omega )\), the statement of the preceding lemma reduces to the usual convergence assertion in the \(L^{m+1}\)-sense,
2.5 An energy bound
In this section, we derive an energy bound for variational solutions.
Lemma 2.20
(Energy bound) Assume that (1.9) and (1.10) are satisfied and that u is a variational solution to (1.2) in the sense of Definition1.1. Then, u fulfills the energy bound
with a constant \(c=c(\ell ,m)\) for any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\).
Proof
If u is a variational solution and v an admissible comparison map in the sense of Definition 1.1, we have that
for a.e. \(\tau \in [0,T]\). By Lemmas 2.3 and 2.6, we obtain that
holds true for any \(\tau \in [0,T]\). For \(\varepsilon >0\), by the generalized Hölder’s inequality (1.8), Hölder’s inequality, Lemma 2.5 applied to \(\Vert \cdot \Vert '_{L^{\Phi ^*}(\Omega )}\) and Young’s inequality, we infer
Inserting (2.13) and (2.14) into (2.12), we conclude that
At this stage, we choose \(\varepsilon := \frac{\ell }{6m}\). 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. \(\square\)
2.6 Compactness
The proof of following result can be found in [11, Proposition 3.1].
Lemma 2.21
Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain, \(p>1\), \(T>0\) and \(k \in \mathbb {N}\). Suppose that for \(h_k := T/k\) piecewise constant maps \(u^{(k)} :\Omega \times (-h_k,T] \rightarrow \mathbb {R}_{\ge 0}\) are defined by
with nonnegative functions \(u_i^{(k)} \in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\). Suppose further that there exists a constant \(M>0\) such that the energy estimate
and the continuity estimate
hold true for all \(k \in \mathbb {N}\), and that \(u^{(k)} \rightharpoondown u\) weakly in \(L^p(0,T;W^{1,p}(\Omega ))\) as \(k \rightarrow \infty\). Then, there exists a subsequence \({\mathfrak {K}} \subset \mathbb {N}\) such that in the limit \({\mathfrak {K}} \ni k \rightarrow \infty\) we have the following convergences:
3 Existence for regular data
First, we prove an existence result for the case of regular boundary values and obstacle. More precisely, we consider nonnegative regular boundary values given by
with a constant \(\kappa >0\) and a nonnegative obstacle function \(\psi\) satisfying
Theorem 3.1
Assume that the obstacle satisfies (3.2) and boundary values are given by (3.1). Then, there exists a variational solution to (1.2) in the sense of Definition 1.1.
3.1 A sequence of minimizers to elliptic variational functionals
Fix a step size \(h \in (0,1]\) such that \(h = T/K\) for some \(K \in \mathbb {N}\). For \(i \in \{0,\ldots ,K\}\) define \(g_i := g(ih)\) and \(\psi _i := \psi (ih)\). Set \(u_0 = g_o\). Then, we inductively find minimizers \(u_i\) of the functionals
in the class \(v \in L^\Phi (\Omega ) \cap \big (g_i+W^{1,p}_0(\Omega )\big )\) with \(v \ge \psi _i\). 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.
Proposition 3.2
Assume that nonnegative functions \(g_*\in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\), \(\psi _*\in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\) and \(u_*\in L^\Phi (\Omega )\) are given. Then, there exists a minimizer u of
in the class of functions \(v \in L^\Phi (\Omega ) \cap \big (g_*+W^{1,p}_0(\Omega )\big )\) with \(v \ge \psi _*\).
Proof
Consider a minimizing sequence \((u_j)_{j \in \mathbb {N}}\), i.e.
By means of Lemmas 2.3 and 2.6, we find that
Together with (1.3) and the fact that \(h \in (0,1]\), this implies
Hence, by (1.7) the sequence \((u_j)_{j \in \mathbb {N}}\) in bounded in \(L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\). Thus, there exists a (not relabeled) subsequence and a limit map \(u \in L^\Phi (\Omega ) \cap \big (g_*+W^{1,p}_0(\Omega )\big )\) such that
in the limit \(j \rightarrow \infty\). Observe that the obstacle condition \(u \ge \psi _*\) a.e. in \(\Omega\) is preserved. Since F is convex and lower semicontinuous with respect to strong convergence in \(L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\) by means of Fatou’s Lemma, F is also lower semicontinuous with respect to weak convergence in \(L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\), cf. [14, Corollary 3.9]. As a consequence, we find that
which yields the claim. \(\square\)
3.2 Energy estimates
Observe that \(v := u_{i-1} + \psi _i - \psi _{i-1} \ge \psi _i\) 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 \(\mathrm {I}\) and \(\mathrm {II}\) is clear in this context. First, we estimate \(\mathrm {I}\). To this end, using the Lipschitz estimate (1.4), Young’s inequality, the assumption \(h \le 1\) and the coercivity assumption (1.3), we conclude that
holds true with a constant \(c=c(p,n,L,\nu )\). Moreover, we find that
and that
Therefore, we deduce the estimate
with a constant \(c=c(n,p,L,\nu )\). Next, by Lemma 2.7 and (1.5) together with Remark 2.2 and the fact that \(u_{i-1} \ge \psi _{i-1} \ge \kappa\) and \(0<\ell \le m \le 1\), we estimate \(\mathrm {II}\), which leads to
Therefore, by Hölder’s inequality, we conclude that
Combining the estimates for \(\mathrm {I}\) and \(\mathrm {II}\), we obtain that
holds true with a constant \(c=c(\kappa ,b(1),\ell ,m,p,n,L,\nu )\). Summing up (3.3) from \(i=1,\ldots ,k\) for some \(k \in \{1,\ldots ,K\}\) leads to
where we used the short-hand notation
for any \(\tau \in (0,T]\). Reabsorbing \(\sum _{i=1}^{k-1} \int _\Omega f(Du_i) \,\mathrm {d}x\) 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 \(j=1,\ldots ,i\) for \(i=1,\ldots ,k-1\). This yields
Inserting this into (3.4), we obtain that
Since \((1+h)^k \le (1+h)^K = (1+\frac{T}{K})^K \le e^T\), we conclude that
for any \(k \in \mathbb {N}\) with \(kh \le T\). Finally, from Lemma 2.7 and (3.5), we infer the estimate
for some constant \(c=c(\kappa ,b(1),\ell ,m,p,n,L,\nu )\).
3.3 Convergence to a limit map
In the following, we set \(h_K := T/K\) for \(K \in \mathbb {N}\). We define the piecewise constant function \(u^{(K)} :\Omega \times (-h_K,T] \rightarrow \mathbb {R}_{\ge 0}\) by
Analogously, we define \(g^{(K)}\) and \(\psi ^{(K)}\). 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 \((u^{(K)})_{K \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;W^{1,p}(\Omega ))\). Since \(u^{(K)} \in g^{(K)} + L^\infty (0,T;W^{1,p}_0(\Omega ))\), there exists a subsequence \({\mathfrak {K}} \subset \mathbb {N}\) and a limit map \(u \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) such that
in the limit \({\mathfrak {K}} \ni K \rightarrow \infty\). By Lemma 2.7 and the energy estimates (3.5) and (3.6), the assumptions of Lemma 2.21 are satisfied for the sequence \((u^{(K)})_{K \in {\mathfrak {K}}}\). Therefore, choosing another subsequence still denoted by \({\mathfrak {K}}\), we obtain that
in the limit \({\mathfrak {K}} \ni K \rightarrow \infty\). At this stage, observe that \(\psi ^{(K)} \rightarrow \psi\) a.e. in \(\Omega _T\) as \(k \rightarrow \infty\). Combining this with (3.8)\(_2\) and the fact that \(u^{(K)} \ge \psi ^{(K)}\) a.e. in \(\Omega _T\), we deduce that u satisfies the obstacle condition \(u \ge \psi\) a.e. in \(\Omega _T\). Next, by means of Lemma 2.7 and (3.5), we conclude that
Thus, we find a subsequence such that \(\Delta _{h_K} \sqrt{\Phi \big (u^{(K)}\big )} \rightharpoondown w\) weakly in \(L^2(\Omega _T)\). In order to characterize w, we use this fact together with (3.8)\(_1\). More precisely, we obtain for any \(\varphi \in C^\infty _0(\Omega _T)\) that
By a density argument, this ensures that \(w = \partial _t \sqrt{\Phi (u)}\) and in particular \(\partial _t \sqrt{\Phi (u)} \in L^2(\Omega _T)\). Hence, we have that \(\partial _t \Phi (u) = 2\sqrt{\Phi (u)}\partial _t \sqrt{\Phi (u)} \in L^1(\Omega _T)\). This implies that \(\Phi (u) \in C^0([0,T];L^1(\Omega ))\) and hence by means of Lemma 2.9 that \(u \in C^0([0,T];L^\Phi (\Omega ))\).
3.4 Minimizing property of the approximations
Observe that for each \(K \in \mathbb {N}\), the map \(u^{(K)}\) is a minimizer of the functional
in the class of functions \(v \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) satisfying \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _T\). Indeed, using the definitions of \({\mathbf {F}}^{(K)}\) 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 \(v \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) with \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _T\) and any \(s \in (0,1)\) the convex combination \(w_s := u^{(K)} + s(v-u^{(K)})\) of \(u^{(K)}\) and v is still admissible, since \(\psi ^{(K)} \le w_s \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\). Then, the minimality of \(u^{(K)}\) and the convexity of f imply that
for any \(s \in (0,1)\), with equality for \(s=0\). Reabsorbing \(\iint _{\Omega _T} (1-s) f\big (Du^{(K)}\big ) \,\mathrm {d}x\mathrm {d}t\) into the left-hand side of the preceding inequality and dividing by s, we find that
holds true. Note that the map \(s \mapsto \tfrac{1}{s}\big (\Phi (w_s) - \Phi \big (u^{(K)}\big )\big )\) is monotone and converges a.e. in \(\Omega _T\) to the \(L^1(\Omega _T)\)-function \(b\big (u^{(K)}\big ) \big (v-u^{(K)}\big )\), since \(\Phi\) is convex. Passing to the limit \(s \downarrow 0\) in the preceding inequality with the aid of the dominated convergence theorem, we deduce that
for any map \(v \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) with \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _T\). Note that in particular \(u^{(K)}(t) = g_o\) for \(t \in (-h_K,0]\). Using the comparison map \(\chi _{(0,T)} v + \chi _{[\tau ,T]} u^{(K)}\) for some \(\tau \in (0,T]\) instead of v, we infer the localized variational inequality
for any \(\tau \in (0,T]\) and \(v \in L^\Phi (\Omega _\tau ) \cap \big (g^{(K)} + L^p(0,\tau ;W^{1,p}_0(\Omega ))\big )\) with \(v \ge \psi ^{(K)}\) a.e. in \(\Omega _\tau\).
3.5 Variational inequality for the limit map
Here, we pass to the limit \(K \rightarrow \infty\) in (3.9) in order to deduce the variational inequality for the limit map. To this end, we consider an arbitrary map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). We extend v to negative times by setting \(v(t) = v(0) \in L^\Phi (\Omega )\) for \(t<0\). Observe that the function \(v_K := v + \psi ^{(K)} - \psi\) is an admissible comparison map in (3.9), since \(v_k \ge \psi ^{(K)}\) and \(v_K \in L^\Phi (\Omega _T) \cap \big (g^{(K)} + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) 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 \(D\psi ^{(K)} \rightarrow D\psi\) in \(L^p(\Omega _T,\mathbb {R}^n)\), we conclude that
in the limit \(K \rightarrow \infty\). Next, shifting the difference quotient in the last term on the right-hand side from \(b\big (u^{(K)}\big )\) to \(\psi ^{(K)}-\psi\), we obtain that
where the definition of \(\mathrm {I}_K - \mathrm {III}_K\) 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 \(\big (b\big (u^{(K)}(t)\big )_{K \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Further, by Lemma 2.13 and since \(\Delta _{h_K} v \rightarrow \partial _t v\) in \(L^1(0,T;L^\Phi (\Omega ))\) as \(K \rightarrow \infty\), we infer
Next, by the generalized Hölder’s inequality (1.8), Lemma 2.12, the definition of \(\psi ^{(K)}\) and since \(\psi \in C^0([0,T];L^\Phi (\Omega ))\), we deduce that
in the limit \(K \rightarrow \infty\). 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 \(\varvec{\delta }_1(h_K)\) and \(\varvec{\delta }_2(h_K)\) are given by
For the characterization of \(\mathrm {B}_0(h_K)\) and \(\varvec{\delta }_2(h_K)\), we used that \(u^{(K)}(t) = g_o\) and \(v(t)=v(0)\) for \(t \in (-h_K,0]\). Since \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), Lemma 2.14 implies that
Next, we consider the first term on the right-hand side of (3.17). Since we have that \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), we find that \(\Delta _{h_K} v \rightarrow \partial _t v\) strongly in \(L^1(0,T;L^\Phi (\Omega ))\). Further, since \(\big (b\big (u^{(K)}(t)\big )\big )_{K \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) and by (3.8)\(_2\), we know that \(b\big (u^{(K)}\big ) \mathop {\rightharpoondown }\limits ^{*}b(u)\) weakly\(^*\) in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Therefore,
Since we are not allowed to pass to the limit \({\mathfrak {K}} \ni K \rightarrow \infty\) in \(\mathrm {B}_\tau (h_K)\) directly, we integrate (3.10) over \(\tau \in (t_o,t_o+\delta )\) for some \(\delta \in (0,T)\) and \(t_o \in [0,T-\delta ]\) and divide the result by \(\delta\). Combining this with (3.17), we find that
Note that \(u^{(K)} \rightarrow u\) a.e. in \(\Omega _T\). Since \(v \in C^0([0,T];L^\Phi (\Omega ))\), we have that \(v(t+h_K) \rightarrow v(t)\) as \({\mathfrak {K}} \ni K \rightarrow \infty\) a.e. in \(\Omega _T\). Since \({\mathfrak {b}}\) is nonnegative, we may apply Fatou’s lemma, which yields
Collecting (3.11), (3.12), (3.14), (3.15), (3.16), (3.19) and (3.21), we infer
Now, we pass to the limit \(\delta \downarrow 0\). Observe that the integrals in the first line of the preceding inequality depend continuously on time by the absolute continuity of the integral. For the boundary term, by the fact that \(\Phi (u(t)) \ge 0\), \(b(u)v \ge 0\) and by Lemma 2.3, we find the dominating function
Since \(u,v \in C^0([0,T];L^\Phi (\Omega ))\) (see Lemma 2.10 for v), the right-hand side of the preceding inequality depends continuously on time. Therefore, using the dominated convergence theorem, we deduce that
Altogether, passing to the limit \(\delta \downarrow 0\), we conclude that
holds true for a.e. \(t_o \in [0,T]\) and any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Thus, u is a variational solution to (1.2) in the sense of Definition 1.1. \(\square\)
4 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 \(0 \le \psi \le g\) a.e. on \(\Omega _T\),
and
For the proof of the desired result, we need the following lemma, cf. [12, Lemma 4.3].
Lemma 4.1
Assume that \(\Omega \subset \mathbb {R}^n\) is a bounded open set with Lipschitz boundary and let \(\Omega _\varepsilon := \{x \in \Omega :{{\,\mathrm{dist}\,}}(x,\partial \Omega ) > \varepsilon \}\) for any \(\varepsilon >0\). Then, for any \(u \in L^p(0,T;W^{1,p}_0(\Omega ))\) we have
Theorem 4.2
Assume that hypotheses (4.1) and (4.2) are satisfies. Then, there exists a variational solution to (1.2) in the sense of Definition 1.1.
The proof of 4.2 is divided into several steps.
4.1 Approximation
Let \(\varepsilon >0\), extend g to \(\mathbb {R}^n \times (0,T)\) by zero and define \(g_\varepsilon := g *\eta _\varepsilon + \varepsilon\), where \(\eta _\varepsilon\) denotes the standard mollifier in \(\mathbb {R}^n\). Then, we have that \(g_\varepsilon \in L^p(0,T;W^{1,p}(\Omega ))\), \(g_{o,\varepsilon } := g_\varepsilon (0) \in C^\infty (\mathbb {R}^n) \subset \big (L^2(\Omega ) \cap W^{1,p}(\Omega )\big )\) and
In order to define mollifications of the obstacle function and comparison maps, we consider the cutoff function \(\zeta _\varepsilon \in W^{1,\infty }(\Omega ,\mathbb {R}_{\ge 0})\) with \(\zeta _\varepsilon \equiv 0\) in \(\Omega \setminus \Omega _\varepsilon\), \(\zeta _\varepsilon \equiv 1\) in \(\Omega _{2\varepsilon }\) and
Then, we define
Note that
Since \({{\,\mathrm{spt}\,}}(\zeta _\varepsilon ) = \overline{\Omega _\varepsilon }\), we conclude that \(\psi _\varepsilon \in g_\varepsilon + L^p(0,T;W^{1,p}_0(\Omega ))\) and \(\psi _\varepsilon (0) \in L^2(\Omega ) \cap \big (g_{o,\varepsilon } + W^{1,p}_0(\Omega )\big )\). Further, since \(\zeta _\varepsilon\) and \(\eta _\varepsilon\) are independent of time, we have that
and
Finally, by a standard property of the applied mollification procedure, we find that \(0<\varepsilon \le \psi _\varepsilon \le g_\varepsilon\). More generally, for any comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\), we define the mollification
Then, we obtain that \(v_\varepsilon \in g_\varepsilon + L^p(0,T;W^{1,p}_0(\Omega ))\), \(\partial _t v_\varepsilon \in L^1(0,T;L^\Phi (\Omega ))\), \(v_\varepsilon (0) \in L^\Phi (\Omega )\) and \(v_\varepsilon \ge \psi _\varepsilon\) a.e. in \(\Omega _T\). Next, we prove that
Indeed, we conclude that
in the limit \(\varepsilon \downarrow 0\). Further, (4.3) in particular implies that there exists a (not relabeled) subsequence such that \(v_\varepsilon \rightarrow v\) a.e in \(\Omega _T\). A similar computation shows that
Moreover, observe that
Hence, by Remark 2.4 the sequence \((b(v_\varepsilon ))_{\varepsilon >0}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Together, the preceding considerations prove that there exists another subsequence such that
Next, we compute that
in the limit \(\varepsilon \downarrow 0\), which yields
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 \(\varepsilon \downarrow 0\). For the first term, by definition of \(\zeta _\varepsilon\) and Lemma 4.1 we conclude that
in the limit \(\varepsilon \downarrow 0\). This proves (4.8).
4.2 Solutions of the regularized problem
For any \(\varepsilon >0\), the mollifications \(g_\varepsilon\) and \(\psi _\varepsilon\) satisfy the assumptions of Theorem 3.1. Hence, there exist corresponding variational solutions \(u_\varepsilon \in C^0([0,T];L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega )) \big)\). Applying Lemma 2.20 with the admissible comparison map \(g_\varepsilon\), we obtain the energy bound
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 \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega )) \big)\) such that
as \(\varepsilon \downarrow 0\). By (4.3) applied to \(v=\psi\) and (4.10)\(_2\), we obtain that
Hence, the obstacle condition \(u \ge \psi\) is satisfied.
4.3 Improved convergence of the solutions
Next, we need to establish
Since \((b(u_\varepsilon ))_\varepsilon\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) by (4.10) and Remark 2.4, there exists a subsequence such that
for some limit map \(w \in L^\infty (0,T;L^{\Phi ^*}(\Omega ))\). Therefore, it remains to prove that w has the structure b(u). To this end, for \(h>0\) we consider mollifications \([u_\varepsilon -\psi _\varepsilon ]_h\) and \([u-\psi ]_h\) according to (2.7) with zero initial values and define
Since \(L^\Phi (\Omega )\) is separable, by Lemma 2.15 we obtain that \(w_{\varepsilon ,h}, w_h \in L^\infty (0,T;L^\Phi (\Omega ))\). Further, we find that \(w_{\varepsilon ,h} \in g_\varepsilon + L^p(0,T;W^{1,p}_0(\Omega ))\) and \(w_h \in g + L^p(0,T;W^{1,p}_0(\Omega ))\). Since \(u_\varepsilon \ge \psi _\varepsilon\), we have that \(w_{\varepsilon ,h} \ge \psi _\varepsilon\). Next, note that (2.6) implies
Thus, by (4.10)\(_2\) and since \(\psi _\varepsilon \rightarrow \psi\) in \(L^p(0,T;W^{1,p}(\Omega ))\), we deduce that for any fixed \(h>0\) the sequence \((\partial _t [u_\varepsilon -\psi _\varepsilon ]_h)_\varepsilon\) is bounded in \(L^p(\Omega _T)\). 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 \(w_{\varepsilon ,h}\) as comparison map in the variational inequality satisfied by \(u_\varepsilon\). Discarding the nonnegative terms \(\iint _{\Omega _T} f(Du_\varepsilon ) \,\mathrm {d}x\mathrm {d}t\) and \({\mathfrak {B}}[u_\varepsilon (T),w_{\varepsilon ,h}(T)]\), we deduce that
which is equivalent to
where the definition of \(\mathrm {I}_h\), \(\mathrm {II}_h\) and \(\mathrm {III}\) is clear in this context. By the generalized Hölder’s inequality (1.8), Hölder’s inequality, Lemmas 2.3 and 2.5, we infer
Note that \(\sup _{t \in [0,T]} \varrho _\Omega (u_\varepsilon (t)) \le 2C\) and \((\Vert \partial _t \psi _\varepsilon \Vert _{L^1(0,T;L^\Phi (\Omega ))})_\varepsilon\) is bounded by (4.7). Further, by Lemma 2.15, (4.5) and (4.10), we find that
Thus, taking into account (1.7), we conclude that
for a constant c independent of \(\varepsilon\) and h. Next, by the growth condition (1.3), Lemma 2.15, (4.8) and (4.9), we obtain that
Finally, by (4.4) applied to \(\psi (0)\) and \(g_o\), we have that \({\mathfrak {b}}[g_{o,\varepsilon },\psi _\varepsilon (0)] \rightarrow {\mathfrak {b}}[g_o,\psi (0)]\) a.e. in \(\Omega _T\) as \(\varepsilon \downarrow 0\). Further, by Lemma 2.3 and since \(b(g_{o,\varepsilon }),\psi _\varepsilon (0) \ge 0\) by Lemma 2.11, we find the dominating function
in \(L^1(\Omega )\) as \(\varepsilon \downarrow 0\). Thus, we conclude that \({\mathfrak {B}}[g_{o,\varepsilon },\psi _\varepsilon (0)] \rightarrow {\mathfrak {B}}[g_o,\psi (0)]\) in the limit \(\varepsilon \downarrow 0\). In particular, this implies
Inserting (4.16), (4.17) and (4.18) into (4.15), we infer
where the constant c is independent of \(\varepsilon\) and h. By Lemma 2.9, Hölder’s inequality and Lemma 2.7, we bound the left-hand side of the preceding inequality from below,
with a constant \(c=c(\ell ,m)\). Since we have already shown that \(\iint _{\Omega _T} \Phi (w_{\varepsilon ,h}) \,\mathrm {d}x\mathrm {d}t\) and \(\iint _{\Omega _T} \Phi (u_\varepsilon ) \,\mathrm {d}x\mathrm {d}t\) stay bounded in the limits \(\varepsilon \downarrow 0\) and \(h \downarrow 0\), we deduce that
where the constant c does not depend on \(\varepsilon\) or h. Further, by (1.6) and Lemma 2.8 for any \(\delta >0\), there exists \(h_o>0\) such that
holds true for any \(0<h<h_o\) and any \(\varepsilon >0\). Choosing \(h_o\) smaller if necessary, by Lemma 2.15 we may assume that
Let \(q := \min \{\ell +1,p\}\). Then, by (4.19) and (4.20) with a suitable choice of \(h>0\) and (4.14), we infer
Since \(\delta\) was arbitrary, this leads to
Passing to a (not relabeled) subsequence, we have that \(u_\varepsilon \rightarrow u\) a.e. in \(\Omega _T\) as \(\varepsilon \downarrow 0\). Together with (4.12), this implies (4.11).
4.4 Passage to the limit
Consider a comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Define mollifications \(v_\varepsilon\) as in Sect. 4.1. Because of the boundary term, we are not allowed to pass to the limit \(\varepsilon \downarrow 0\) in the variational inequality satisfied by \(u_\varepsilon\) and \(v_\varepsilon\) directly. Instead, we integrate over \(\tau \in (t_o,t_o+\delta )\) for some \(\delta \in (0,T)\) and \(t_o \in (0,T-\delta ]\). This leads to
Since f is convex and satisfies the coercivity condition (1.3)\(_1\), by (4.10)\(_2\) we have that
Further, note that
By (1.6), the functionals \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi (\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) and \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi ^*(\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) are continuous with respect to strong convergence in \(L^\Phi (\Omega _T)\), respectively \(L^{\Phi ^*}(\Omega _T)\). Together with the convexity of \(\Phi\) and \(\Phi ^*\), this implies that they are lower semicontinuous with respect to weak convergence in \(L^\Phi (\Omega _T)\), respectively \(L^{\Phi ^*}(\Omega _T)\), cf. [14, Corollary 3.9]. Therefore, by (4.3) and (4.11), we conclude that
Next, by the local Lipschitz condition (1.4) and (4.8), we find that
From (4.6), (4.7), (4.11) and the dominated convergence theorem, we infer
Finally, by (4.4) applied to v and g and the dominated convergence theorem, we have that
Inserting (4.22), (4.23), (4.24), (4.25) and (4.26) into (4.21), we deduce that
holds true for any \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Passing to the limit \(\delta \downarrow 0\), we conclude that u is a variational solution to (1.2) in the sense of Definition 1.1. \(\square\)
5 Proof of Theorem 1.2
5.1 Regularization
We consider a sequence \(0< h_i \downarrow 0\) as \(i \rightarrow \infty\) and set \(\varepsilon _i := h_i^\frac{1}{2(n+1)p}\). Then, we extend the initial datum \(g_o\) by zero to a function \(g_o \in L^\Phi (\mathbb {R}^n)\) and define mollifications of the extension by
where \(\eta _\varepsilon\) denotes a standard mollifier in \(\mathbb {R}^n\). Thus, we have that \(g_{o,i} \in L^\Phi (\Omega ) \cap W^{1,p}(\Omega )\). Further, we set
where \([g]_{h_i}\) denotes the mollification in time according to (2.7) with initial values \(g_{o,i}\). Since the space \(L^\Phi (\Omega )\) is separable, Lemma 2.15 implies \(g_i \in C^0([0,T];L^\Phi (\Omega )) \cap L^p(0,T;W^{1,p}(\Omega ))\). Moreover, by (2.6) we obtain
Next, we define mollifications of comparison maps \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). To this end, we set
where \(\chi\) is the indicator function, i.e., for \(A \subset \mathbb {R}^n\) we have that \(\chi _A \equiv 1\) in A and \(\chi _A \equiv 0\) in \(\mathbb {R}^n \setminus A\). By definition of \(\eta _\varepsilon\), we obtain that \(v_{o,i} \in L^\Phi (\Omega ) \cap \big (g_{o,i} + W^{1,p}_0(\Omega )\big )\). Then, by
we denote the mollification given by (2.7) with initial datum \(v_{o,i}\). Lemma 2.15 implies that \(v_i \in C^0([0,T];L^\Phi (\Omega )) \cap \big (g_i + L^p(0,T;W^{1,p}_0(\Omega ))\big )\). Moreover, by (2.6), we conclude that
In particular, we apply the preceding mollification procedure to \(v=\psi\). Since \(\psi _{o,i} \le g_{o,i}\) by Lemma 2.11 and the definition of \(\eta _\varepsilon\), we have that \(0 \le \psi _i \le g_i\). In the following, we will also use the mollification
according to (2.7) with \(v_o = v(0)\). In particular, we know that \({\tilde{v}}_i = v_i + e^{-\frac{t}{h_i}}(v(0)-v_{o,i})\). First, we find that
Then, we claim that
Indeed, from Lemma 2.15, we conclude that \({\tilde{v}}_i \rightarrow v\) in \(L^\infty (0,T;L^\Phi (\Omega ))\) as \(i \rightarrow \infty\). Together with (5.1), we obtain that
Moreover, from Lemma 2.3, (1.7) and (5.2), we conclude that
Together with (5.2), this implies that
Next, we show that
To this end, from Lemma 2.16, we infer
i.e., the mollification of \(\partial _t v\) according to (2.7) with zero initial datum. Therefore, by Lemma 2.15, we obtain that \(\partial _t {\tilde{v}}_i \rightarrow \partial _t v\) in \(L^1(0,T;L^\Phi (\Omega ))\) as \(i \rightarrow \infty\), which allows us to compute
Finally, we establish the assertion
To this end, we consider the mollification according to (2.7) with zero initial values, i.e., \(v_i^o := v_i -e^{-\frac{t}{h_i}} v_{o,i}\). Lemma 2.15 implies
Further, observe that
Using Young’s inequality for convolutions, we estimate the last integral. This yields
with \(c=c(n,p,m,|\Omega |)\). Joining the two preceding estimates and recalling the definition of \(\varepsilon _i\), we obtain that
in the limit \(i \rightarrow \infty\). Finally, combining the preceding assertion with (5.6), we obtain (5.5).
5.2 Solutions corresponding to the approximations
As we have shown in the previous section, the approximations \(g_i\) and \(\psi _i\) satisfy the assumptions of Theorem 4.2. Consequently, for each \(i \in \mathbb {N}\) there exists a variational solution \(u_i \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g_i + L^p(0,T;W^{1,p}_0(\Omega ))\big )\) corresponding to \(g_i\) and \(\psi _i\). By Lemma 2.20 with \(v=g_i\) we deduce the energy bound
where the constant C is given by
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 \(u \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}_0(\Omega )) \big)\) and a (not relabeled) subsequence such that
in the limit \(i \rightarrow \infty\). The obstacle condition is preserved as \(i \rightarrow \infty\). Indeed, by (5.8) and (5.2) applied to \(v_i=\psi _i\), we find that
5.3 Convergence of solutions
In this step, we wish to establish
Since \((b(u_i))_{i \in \mathbb {N}}\) is bounded in \(L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) by Lemma 2.3, (1.7) and (5.7), we know that there exists \(w \in L^\infty (0,T;L^{\Phi ^*}(\Omega ))\) such that (for a subsequence)
However, it remains to prove that w has the structure b(u). To this end, let \(\lambda >0\) and consider mollifications \([u_i-\psi _i]_\lambda\) and \([u-\psi ]_\lambda\) according to (2.7) with zero initial datum. Then, we define
By Lemma 2.15 we obtain that \(w_{i,\lambda } \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g_i + L^p(0,T;W^{1,p}(\Omega )) \big)\) and \(w_\lambda \in L^\infty (0,T;L^\Phi (\Omega )) \cap \big (g + L^p(0,T;W^{1,p}(\Omega )) \big)\). Furthermore, (2.6) implies that
Since \(\psi _i \rightarrow \psi\) in \(L^p(\Omega _T)\), by (5.8)\(_2\) and Lemma 2.17, the sequence \(([u_i-\psi _i]_\lambda )_{i \in \mathbb {N}}\) is bounded in \(L^p(\Omega _T)\) for any fixed \(\lambda >0\). Further, by (5.5), (5.8)\(_2\) and Lemma 2.17
holds true for fixed \(\lambda >0\). Therefore, we conclude from Rellich’s theorem that
for any fixed \(\lambda >0\). Here, we did not have to pass to a subsequence, since the limit is determined by (5.11). Since \(u_i \ge \psi _i\) for any \(i \in \mathbb {N}\), we know that \(w_{i,\lambda } \ge \psi _i\). Using \(w_{i,\lambda }\) as comparison map in the variational inequality for \(u_i\) leads us to
The preceding inequality is equivalent to
where the definition of \(\mathrm {I}_\lambda\), \(\mathrm {II}_\lambda\) and \(\mathrm {III}\) 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 \(v_i=\psi _i\), the first factor on the right-hand side of the preceding inequality stays bounded in the limit \(i \rightarrow \infty\). Further, by the energy bound (5.7) we know that \(\sup _{t \in [0,T]}\int _\Omega \Phi (u_i) \,\mathrm {d}x\le 2C\) for any \(i \in \mathbb {N}\). By Lemma 2.15, (5.2) and (5.8)\(_1\) we find that
Consequently, by (1.7), we conclude that \(\sup _{t \in [0,T]}\int _\Omega \Phi (w_{i,\lambda }) \,\mathrm {d}x\) is bounded by a constant independent of \(i \in \mathbb {N}\) and \(\lambda >0\). Altogether, we obtain that
with a constant c independent of i and \(\lambda\). Next, by the growth condition (1.3), Lemma 2.15 and (5.5), we deduce that
Finally, by (5.1), we have that \({\mathfrak {b}}[g_{o,i},\psi _i(0)] \rightarrow {\mathfrak {b}}[g_o,\psi (0)]\) a.e. in \(\Omega _T\). Further, by Lemma 2.3, the nonnegativity of \(b(g_{o,i})\) and \(\psi _i(0)\) and (5.1), we find the dominating function
Consequently, by a version of the dominated convergence theorem (cf. [16, Theorem 1.20]), we know that \({\mathfrak {B}}[g_{o,i},\psi _i(0)] \rightarrow {\mathfrak {B}}[g_o,\psi (0)]\) as \(i \rightarrow \infty\) and hence
Inserting (5.14), (5.15) and (5.16) into (5.13), we obtain
with a constant c independent of \(i \in \mathbb {N}\) and \(\lambda >0\). In order to bound the left-hand side of the preceding inequality from below, we apply Lemma 2.9, Hölder’s inequality and Lemma 2.7. This yields
We have already established that the first factor on the right-hand side of the preceding inequality is bounded independent of i and \(\lambda\). Therefore, we obtain that
where the constant is independent of i and \(\lambda\). Thus, by (1.6) and Lemma 2.8 for any \(\delta >0\), there exists \(\lambda _o>0\) such that
holds true for any \(0<\lambda <\lambda _o\). Decreasing \(\lambda _o\) if necessary, by Lemma 2.15, we may assume that
holds true for any \(0<\lambda <\lambda _o\) as well. Finally, we abbreviate \(q := \min \{p,\ell +1\}\). Combining (5.12), (5.17) and (5.18) with the choice \(0<\lambda <\lambda _o\), we infer
Since \(\delta >0\) was arbitrary, the preceding consideration yields
Passing to a subsequence, we further have that \(u_i \rightarrow u\) a.e. in \(\Omega _T\) as \(i \rightarrow \infty\). Together with (5.10), this proves (5.9).
5.4 Conclusion of the proof
Consider a comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). We define mollifications \(v_i\) to v as in Sect. 5.1. Since we are not able to pass to the limit \(i \rightarrow \infty\) in the variational inequality satisfied by \(u_i\) and \(v_i\) directly, we integrate over \(\tau \in (t_o,t_o+\delta )\) for some \(\delta \in (0,T)\) and \(t_o \in (0,T-\delta ]\). This leads to
In the following, we consider the terms of (5.19) separately. Since f is convex and satisfies the growth condition (1.3), we conclude from (5.8)\(_2\) that
In order to treat the second term on the left-hand side of (5.19), note that
Since \(\Phi\) is convex and the functional \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi (\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) is continuous with respect to strong convergence in \(L^\Phi (\Omega _T)\) by (1.6), we conclude that it is lower semicontinuous with respect to weak convergence in \(L^\Phi (\Omega _T)\), cf. [14, Corollary 3.9]. Similarly, we find that \(\mathop {\int \!\!\!\!\!\!-}\nolimits _{t_o}^{t_o+\delta } \int _\Omega \Phi ^*(\cdot ) \,\mathrm {d}x\mathrm {d}\tau\) is lower semicontinuous with respect to weak convergence in \(L^{\Phi ^*}(\Omega _T)\). Thus, as a consequence of (5.2) and (5.9), we obtain that
By the Lipschitz condition (1.4) and (5.5), we have that
Next, by (5.3), (5.4) and (5.8)\(_1\), we conclude that
Finally, by (5.1) applied to \(g_{o,i}\) and \(v_i(0)\) and the dominated convergence theorem, we infer
Collecting (5.20), (5.21), (5.22), (5.23) and (5.24), we deduce from (5.19) that
holds true for any admissible comparison map \(v \in g + L^p(0,T;W^{1,p}_0(\Omega ))\) with \(\partial _t v \in L^1(0,T;L^\Phi (\Omega ))\), \(v(0) \in L^\Phi (\Omega )\) and \(v \ge \psi\) a.e. in \(\Omega _T\). Passing to the limit \(\delta \downarrow 0\), we conclude that u is a variational solution to (1.2) in the sense of Definition 1.1. \(\square\)
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, London (2003)
Akagi, G., Stefanelli, U.: Doubly nonlinear equations as convex minimization. SIAM J. Math. Anal. 46(3), 1922–1945 (2014)
Alt, H., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183(3), 311–341 (1983)
Aronson, D.G.: Regularity properties of flows through porous media: the interface. Arch. Ration. Mech. Anal. 37, 1–10 (1970)
Barenblatt, G.I.: On some unsteady motions of a liquid and gas in a porous medium. Akad. Nauk SSSR. Prikl. Mat. Meh. 16, 67–78 (1952). (Russian)
Barenblatt, G.I.: On self-similar solutions of the Cauchy problem for a nonlinear parabolic equation of unsteady filtration of a gas in a porous medium. Prikl. Mat. Meh. 20, 761–763 (1956). (Russian)
Bernis, F.: Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. Math. Ann. 279(3), 373–394 (1988)
Bögelein, V., Duzaar, F., Marcellini, P.: A time dependent variational approach to image restoration. SIAM J. Imaging Sci. 8(2), 968–1006 (2015)
Bögelein, V., Duzaar, F., Marcellini, P.: Existence of evolutionary variational solutions via the calculus of variations. J. Differ. Equ. 256, 3912–3942 (2014)
Bögelein, V., Duzaar, F., Marcellini, P.: Parabolic systems with \(p, q\)-growth: a variational approach. Arch. Ration. Mech. Anal. 210(1), 219–267 (2013)
Bögelein, V., Duzaar, F., Marcellini, P., Scheven, C.: Doubly nonlinear equations of porous medium type. Arch. Ration. Mech. Anal. 229, 503–545 (2018)
Bögelein, V., Duzaar, F., Scheven, C.: The obstacle problem for parabolic minimizers. J. Evol. Equ. 17(4), 1273–1310 (2017)
Bögelein, V., Lukkari, T., Scheven, C.: The obstacle problem for the porous medium equation. Math. Ann. 363(1–2), 455–499 (2015)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)
Dal Passo, R., Luckhaus, S.: A degenerate diffusion problem not in divergence form. J. Differ. Equ. 69(1), 1–14 (1987)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, Revised edn. Chapman and Hall/CRC, Boca Raton (2015)
Grange, O., Mignot, F.: Sur la résolution d’une équation et d’une inéquation paraboliques non linéaires. J. Funct. Anal. 11, 77–92 (1972). (French)
Ivanov, A.V.: Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83(1), 22–37 (1997)
Ivanov, A.V., Mkrtychyan, P.Z.: On the existence of Hölder-continuous generalized solutions of the first boundary value problem for quasilinear doubly degenerate parabolic equations. J. Sov. Math. 62(3), 2725–2740 (1992)
Ivanov, A.V., Mkrtychyan, P.Z., Jäger, W.: Existence and uniqueness of a regular solution of the Cauchy–Dirichlet problem for a class of doubly nonlinear parabolic equations. J. Sov. Math. 84(1), 845–855 (1997)
Ladyženskaja, O.A.: New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. Trudy Mat. Inst. Steklov. 102, 85–104 (1967). (Russian)
Landes, R.: On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc. R. Soc. Edinb. Sect. A 89(3–4), 217–237 (1981)
Lichnewsky, A., Temam, R.: Pseudosolutions of the time-dependent minimal surface problem. J. Differ. Equ. 30(3), 340–364 (1978)
Marcellini, P.: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscr. Math. 51, 1–28 (1985)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces, Pure and Applied Mathematics. Marcel Dekker, New York (1991)
Schätzler, L.: Existence for singular doubly nonlinear systems of porous medium type with time dependent boundary values. J. Elliptic Parabol. Equ. 5, 383–421 (2019)
Schätzler, L.: The obstacle problem for singular doubly nonlinear equations of porous medium type. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (to appear) (2020)
Showalter, R., Walkington, N.J.: Diffusion of fluid in a fissured medium with microstructure. SIAM J. Math. Anal. 22(6), 1702–1722 (1991)
Acknowledgements
Open Access funding provided by Projekt DEAL.
Funding
L. Schätzler has been supported by Studienstiftung des deutschen Volkes.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
No conflicts of interest have occurred.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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/.
About this article
Cite this article
Schätzler, L. The obstacle problem for degenerate doubly nonlinear equations of porous medium type. Annali di Matematica 200, 641–683 (2021). https://doi.org/10.1007/s10231-020-01008-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-020-01008-y