Non-local multi-class traffic flow models

To Giuseppe Mingione, on the occasion of his 50th birthday, with regard and admiration.

1.   Introduction

The aim of this paper is to study a nonlinear and noncoercive parabolic variational inequality with constraint and homogeneous Dirichlet boundary condition. The Lewy-Stampacchia inequality associated with it is addressed. After the first results of H. Lewy and G. Stampacchia ^[19] concerning inequalities in the context of superharmonic problems, there is by now a large literature concerning the theory of elliptic obstacle problems as well as of elliptic variational inequalities. We refer to ^[3,16,25] for a classical overview. For a more recent treatment related to nonlinear elliptic operators see also ^[23]. The obstacle problem for nonlocal and nonlinear operators has been cosidered in ^[17,26]. An abstract and general version of the Lewy-Stampacchia inequality is given in ^[13]. Concerning the parabolic case, first existence results related to problems with time independent obstacles have been treated in ^[20] in the linear case and in ^[5] for the more general parabolic problems. The case of obstacles functions regular in time has been considered in ^[2,5]. Existence and regularity theory for solutions of parabolic inequalities involving degenerate operators in divergence form have been established in ^[4,18]. More recently in ^[15], the Authors prove Lewy-Stampacchia inequality for parabolic problems related to pseudomonotone type operators. In this paper we study a variational parabolic inequality for noncoercive operators that present singularities in the coeffcients of the lower order terms in the same spirit of ^[9,12,14].

Let us state the functional setting and the assumptions on the data.

Let $ \Omega\subset \mathbb R^N $, $ N \geqslant2 $, be a bounded open Lipschitz domain and let $ \Omega_T: = \Omega \times (0, T) $ be the parabolic cylinder over $ \Omega $ of height $ T > 0 $. We shall denote by $ \nabla v $ and $ \partial _ t v $ (or $ v_t $) the spatial gradient and the time derivative of a function $ v $ respectively. We consider the class

where

and $ p^\prime $ is the conjugate exponent of $ p $, i.e., $ \frac 1 p + \frac 1 {p^\prime} = 1 $. In (1.1), $ L^p(0, T, W^{1, p}_0(\Omega)) $ and $ L^{p'}\left(0, T, W^{-1, p'}(\Omega)\right) $ denote parabolic Banach spaces defined according to (2.7).

Given a measurable function $ \psi \colon \, \Omega_T \cup \Omega \times\{0\} \rightarrow \mathbb{R} $, we are interested in finding functions $ u\colon \Omega_T\rightarrow \mathbb R $ in the convex subset $ \mathcal K_{\psi}(\Omega_T) $ of $ W_p(0, T) $ defined as

and satisfying the following variational inequality

where

and $ \langle\cdot, \cdot\rangle $ denotes the duality between $ W^{-1, p'}(\Omega) $ and $ W_0^{1, p}(\Omega) $. The vector field

is a Carathéodory function, i.e., $ A $ measurable w.r.t. $ (x, t)\in \Omega_T $ for all $ (u, \xi)\in \mathbb R \times \mathbb{R}^N $ and continuous w.r.t. $ (u, \xi)\in \mathbb R\times \mathbb{R}^N $ for a.e. $ (x, t)\in \Omega_T $, and such that for a.e. $ (x, t)\in \Omega_T $ and for any $ u\in \mathbb R $ and $ \xi, \eta \in \mathbb{R}^N $,

hold true. Here $ \alpha, \beta $ are positive constants, while $ H $, $ K $, $ b $ and $ \tilde b $ are nonnegative measurable functions defined on $ \Omega_T $ such that $ H \in L^1(\Omega_T) $, $ K\in L^{p^\prime}(\Omega_T) $ and

where $ L^{N, \infty}(\Omega) $ is the Marcinkiewicz space. For definitions of $ L^{N, \infty}(\Omega) $ and $ L^\infty \left(0, T, L^{N, \infty} (\Omega) \right) $ see Sections 2.2 and 2.3, respectively.

We assume that the obstacle function fulfills

For

we impose the following compatibility condition

In the following, we will refer to a function $ u\in \mathcal K_{\psi}(\Omega_T) $ satisfying (1.3) and such that $ u(\cdot, 0) = u_0 $ as a solution to the variational inequality in the strong form with initial value $ u_0 $.

Under previous assumptions the existence of a solution in the weak form can be proved, see ^[12]. However the existence of a solution in the sense stated above is not guaranteed even in simpler cases. Then we assume that the source term and the obstacle function are such that

Here $ L^{p'}(0, T, W^{-1, p'}(\Omega))^+ $ denotes the non-negative elements of $ L^{p'}(0, T, W^{-1, p'} (\Omega)) $. Following the terminology of ^[7] or ^[15], (1.15) is equivalent to say that $ g $ is an element of the order dual $ L^p(0, T, W_0^{1, p}(\Omega))^\ast $ defined as

Then, our main result reads as follows

Theorem 1.1. Let (1.2) and (1.4)–(1.15) be in charge. Assume further that

where $ S_{N, p} = \omega_N^{-1/N}\frac{p }{ N-p } $ and $ \omega_N $ denotes the measure of the unit ball of $ \mathbb{R}^N $. Then, there exists at least a solution $ u \in \mathcal K_{\psi}(\Omega_T) $ of the strong form of the variational inequality (1.3) satisfying $ u(\cdot, 0) = u_0 $. Moreover, the following Lewy-Stampacchia inequality holds

In (1.16), $ \mathscr D_b $ denotes the distance of $ b $ from $ L^\infty(\Omega_T) $ in the space $ L^\infty(0, T, L^{N, \infty}(\Omega)) $ defined in (2.8) below.

Assumptions (1.8) on the coefficients of the lower order terms allow us to consider diffusion models in which the boundedness of the convective field with respect to the spatial variable is too restrictive (see ^[8]). The corresponding bounded case has been treated in ^[15].

We discuss condition (1.16) through an example. It's easy to verify that the operator

satisfies (1.5)–(1.8). According to (2.2) and (2.3) below, we get that

and so (1.16) holds true whenever $ \gamma $ is small enough. On the other hand, we notice that (1.16) does not imply smallness of the norm of the coefficient $ b $. Indeed

for a constant $ C $ independent of $ \gamma $.

Theorem 1.1 also applies in the case $ b $ and $ \tilde b $ lie in a functional subspace of weak–$ L^N $ in which bounded functions are dense. For more details see also ^[10]. For other examples of operators satisfying conditions above we refer to ^[12].

We remark that for $ f, \psi_t, \text{div } A(x, t, \psi, \nabla \psi) \in L^{p^\prime}(\Omega_T) $ condition (1.15) is satisfied. Then, Theorem 1.1 is comparable with the existence result of Lemma 3.1 in ^[4]. In order to prove our result, we consider a sequence of suitable penalization problems for which an existence result holds true (see ^[12]). Then we are able to construct a solution u to (1.3) as limit of solutions of such problems despite the presence of unbounded coefficients in the lower order terms.

2.   Preliminary results

In this section we provide the notation and several preliminary results that will be fundamental in the sequel.

2.1.   Notation

The symbol $ C $ (or $ C_1, C_2, \dots $) will denote positive constant, possibly varying from line to line. For the dependence of $ C $ upon parameters, we will simply write $ C = C(\cdot, \dots, \cdot) $. The positive and the negative part of a real number $ z $ will be denoted by $ z^+ $ and $ z^- $, respectively, and are defined by $ z^+: = \max\{z, 0\} $ and $ z^-: = -\min\{z, 0\} $. Given $ z_1, z_2 \in \mathbb R $, we often use the notation $ z_1 \wedge z_2 $ and $ z_1 \vee z_2 $ in place of $ \min\{z_1, z_2\} $ and $ \max\{z_1, z_2\} $ respectively.

2.2.   Lorentz spaces

Let $ \Omega $ be a bounded domain in $ \mathbb{R}^N $. For any $ 1 < p < \infty $ and $ 1\leq q < \infty $, the Lorentz space $ L^{p, q}(\Omega) $ is the set of real measurable functions $ f $ on $ \Omega $ such that

Here $ \lambda_f(k): = \left| \left\{ x\in \Omega: |f(x)| > k \right\} \right| $ is the distribution function of $ f $. When $ p = q $, the Lorentz space $ L^{p, p}(\Omega) $ coincides with the Lebesgue space $ L^p(\Omega) $. When $ q = \infty $, the space $ L^{p, \infty}(\Omega) $ is the set of measurable functions $ f $ on $ \Omega $ such that

This set coincides with the Marcinkiewicz space weak-$ L^p(\Omega) $. The expressions above do not define a norm in $ L^{p, q} $ or $ L^{p, \infty} $ respectively, in fact triangle inequality generally fails. Nevertheless, they are equivalent to a norm, which make $ L^{p, q}(\Omega) $ and $ L^{p, \infty}(\Omega) $ Banach spaces when endowed with them. An important role in the potential theory is played by these spaces as pointed out in ^[22].

For $ 1\leq q < p < r\leq \infty $, the following inclusions hold

For $ 1 < p < \infty $, $ 1\leq q \leq \infty $ and $ \frac 1 p+\frac 1 {p'} = 1 $, $ \frac 1 q+\frac 1 {q'} = 1 $, if $ f\in L^{p, q}(\Omega) $, $ g\in L^{p', q'}(\Omega) $ we have the Hölder–type inequality

Since $ L^\infty(\Omega) $ is not dense in $ L^{p, \infty}(\Omega) $, for $ f \in L^{p, \infty} (\Omega) $ in ^[6] the Authors stated the following

As already observed in ^[10,11], we have

and

where, for all $ m > 0 $, $ \mathcal T_m $ is the truncation operator at levels $ \pm m $, i.e.,

Another useful estimate is provided by the following sort of triangle inequality

which holds true for $ f, g\in L^{p, \infty}(\Omega) $ and $ \varepsilon > 0 $.

For $ 1\leq q < \infty $, any function in $ L^{p, q}(\Omega) $ has zero distance to $ L^\infty(\Omega) $. Indeed, $ L^\infty(\Omega) $ is dense in $ L^{p, q}(\Omega) $, the latter being continuously embedded into $ L^{p, \infty}(\Omega) $.

Assuming that $ 0\in\Omega $, $ b(x) = \gamma/|x| $ belongs to $ L^{N, \infty}(\Omega) $, $ \gamma > 0 $. For this function, we have

The Sobolev embedding theorem in Lorentz spaces ^[1,24] reads as

Theorem 2.1. Let us assume that $ 1 < p < N $, $ 1\leq q\leq p $, then every function $ u\in W_0^{1, 1}(\Omega) $ verifying $ |\nabla u|\in L^{p, q}(\Omega) $ actually belongs to $ L^{p^*, q}(\Omega) $, where $ p^*: = \frac{Np}{N-p} $ is the Sobolev conjugate exponent of $ p $ and

where $ S_{N, p} $ is the Sobolev constant given by $ S_{N, p} = \omega_N^{-1/N} \frac{p }{ N-p } $.

2.3.   Parabolic spaces

Let $ T > 0 $ and $ X $ be a Banach space endowed with a norm $ \|\cdot\|_X $. Then, the space $ L^p\left(0, T, X\right) $ is defined as the class of all measurable functions $ u\colon [0, T] \rightarrow X $ such that

whenever $ 1\leq p < \infty $, and

for $ p = \infty $. The space $ C^0\left([0, T], X\right) $ represents the class of all continuous functions $ u\colon [0, T] \rightarrow X $ with the norm

We essentially consider the case where $ X $ is either a Lorentz space or Sobolev space $ W^{1, p}_0(\Omega) $. This space will be equipped with the norm $ \|g\| _{W^{1, p}_0(\Omega)} : = \| \nabla g \| _{L^p(\Omega)} $ for $ g\in W^{1, p}_0(\Omega) $.

For $ f\in L^\infty(0, T, L^{p, \infty}(\Omega)) $ we define

and as in (2.3) we find

In the class $ W_p(0, T) $ defined in (1.1) and equipped with the norm

the following inclusion holds (see [27,Chapter III, page 106]).

Lemma 2.2. Let $ p > 2N /(N+2) $. Then $ W_p(0, T) $ is contained into the space $ C^0\left([0, T], L^2(\Omega)\right) $ and any function $ u \in W_p(0, T) $ satisfies

for some constant $ C > 0 $.

Moreover, the function $ t\in[0, T]\mapsto\|u(\cdot, t)\|^2_{L^2(\Omega)} $ is absolutely continuous and

The compactness result due to Aubin–Lions reads as follows.

Lemma 2.3. Let $ X_0, X, X_1 $ be Banach spaces with $ X_0 $ and $ X_1 $ reflexive. Assume that $ X_0 $ is compactly embedded into $ X $ and $ X $ is continuously embedded into $ X_1 $. For $ 1 < p, q < \infty $ let

Then $ W $ is compactly embedded into $ L^p(0, T, X) $.

As an example, we choose $ q = p' $, $ X_0 = W^{1, p}_0(\Omega) $, $ X_1 = W^{-1, p^\prime} (\Omega) $ and $ X = L^p(\Omega) $ if $ p\geq 2 $ or $ X = L^2(\Omega) $ for $ \frac {2N}{N+2} < p < 2 $. Therefore, we deduce

Lemma 2.4. If $ p > 2N /(N+2) $ then $ W_p(0, T) $ is compactly embedded into $ L^p(\Omega_T) $ and into $ L^2(\Omega_T) $.

3.   A penalized problem

Let $ \delta > 0 $. We introduce the following initial–boundary value problem

where

Moreover, in this section we assume that

We introduce the notation

By the elementary inequality

and recalling (1.5), (1.6) and (1.7), we easily deduce

for a.e. $ (x, t)\in \Omega_T $ and for any $ u\in \mathbb R $ and $ \xi, \eta \in \mathbb{R}^N $.

For $ u_0 \in L^2(\Omega) $ and $ f \in L^{p'}(0, T, W^{-1, p'}(\Omega)) $, a solution to problem (3.1) is a function

such that

for every $ \varphi \in C^\infty(\bar \Omega_T) $ such that $ {\rm supp }\, \varphi \subset [0, T)\times \Omega $.

By using the elementary inequality

and Young inequality we see that

Hence, by Theorem 4.2 and Remark 4.5 in ^[12] we get the following existence result.

Proposition 3.1. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. For every fixed $ \delta > 0 $, problem (3.1) admits a solution $ u_{\delta} \in C^0\left([0, T], L^2(\Omega)\right) \cap L^p(0, T, W^{1, p}_0(\Omega)) $.

The arguments of ^[12] lead to some estimates for the sequence $ \{u_\delta\}_{\delta > 0} $. We propose here a proof that carefully keeps trace of the constants in the estimates.

Lemma 3.2. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. Any solution $ u_\delta \in C^0\left([0, T], L^2(\Omega)\right) \cap L^p(0, T, W^{1, p}_0(\Omega)) $ to problem (3.1) satisfies the following estimate

Proof. We fix $ t \in (0, T) $ and we set $ \Omega_t: = \Omega\times (0, t) $. We choose $ \varphi: = \mathcal T_1(u_\delta)\chi_{(0, t)} $ as a test function. If we let $ \Phi(z): = \int_0^z \mathcal T_1(\zeta) \, \mathrm d \zeta $ for $ z\in\mathbb R $, we have

Assumption (3.2) implies that $ [(\psi- u_{\delta})^+]^{q-1} \mathcal T_1(u_\delta) \leqslant 0 $ a.e. in $ \Omega_T $, so we have

By (1.5) and (1.7) we deduce

Now, as $ 0 \leqslant \Phi(z) \leqslant \frac {z^2} 2 $ for all $ z\in \mathbb R $, we have

By Hölder and Young inequality we get

Finally, by (3.3)

Gathering (3.6), (3.7), and (3.8) and using Hölder inequality, by (3.5) we have

where

It is easily seen that

and so

We fix $ t \in (0, T) $ and choose $ \varphi: = u_\delta \chi_{(0, t)} $ as a test function in (3.1). Again, assumption (3.2) implies that $ [(\psi- u_{\delta})^+]^{q-1} u_\delta \leqslant 0 $ a.e. in $ \Omega_T $, then

By Young inequality for $ \varepsilon > 0 $

Then, by (1.5) we further have

For $ m > 0 $ to be chosen later, we have from (3.3)

We estimate separately the two terms in the right–hand side of (3.12). For $ k > 1 $ fixed, we obtain

Now we apply Hölder inequality (2.1), estimates (2.6) and (3.10) to get

where $ M_0 $ is the constant in (3.9). On the other hand, using again Hölder inequality (2.1) and estimate (2.6)

we have

Inserting (3.13), (3.14) and (3.15) into (3.12) we obtain

Observe that (3.11) and (3.16) imply

Now we choose $ m > 0 $ so large to guarantee

The existence of such a value of $ m $ is a direct consequence of (1.16) and the characterization of distance in (2.9). It is also clear that $ m $ is a positive constant depending only on $ b $, $ N $, $ p $ and $ \alpha $. So we get

for some $ \alpha_1 = \alpha_1(b, N, p, \alpha) $. We may also choose $ \varepsilon = \frac{\alpha_1}{2} $. Then the latter relation becomes

We choose $ k = M_0 \left(\frac{\alpha_1}{4 C_2} \right)^{N/p} $ so that $ C_2\left(\frac {M_0} k\right)^{p/N} = \frac {\alpha_1} 4 $ and therefore

Taking into account the definition of $ M_0 $, the latter leads to the estimate (3.4).

Lemma 3.3. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. Assume further that $ g^- $ defined in (1.15) is such that

Then, for every $ \delta > 0 $, every solution $ u_\delta $ of problem (3.1) satisfies

Moreover, there exists a positive constant C depending only on the data and independent on $ \delta $ such that

Proof. We use the function $ \phi = (\psi-u_\delta)^+ $ as a test function in the equation of Problem (3.1). Then, we get

Recalling (1.15), this implies

By (1.14) we observe that

hence, by (1.6) we get

Then, using Hölder inequality and dividing both sides of the inequality by $ \| (\psi- u_\delta)^+)\|_{L^q((\Omega_T)} $ we obtain (3.18). To obtain (3.19) we fix $ \varphi \in L^p(0, T; W_0^{1, p}(\Omega)) $ and then we observe that

At this point we observe that the definition of $ q $ and Holder inequality imply

Finally, using (3.18) and Poncaré inequality slicewise, we conclude that

where $ C $ is a positive constant independent of $ \delta. $ This immediately leads to (3.19).

4.   Proof of main result

We proceed step by step. We first prove the result under regularity assumptions on $ g $ and sign conditon (3.2) on the obstacle function $ \psi $. Then we address the general case.

Proposition 4.1. Let (1.2), (1.4)–(1.16), (3.2) and (3.17) be in charge. There exists at least solution $ u \in \mathcal K_{\psi}(\Omega_T) $ to the variational inequality (1.3) such that $ u(\cdot, 0) = u_0 $ in $ \Omega $ and satisfying the following estimate

Proof. By Proposition 3.1, for every $ \delta > 0 $ there exists a solution $ u_{\delta} \in C^0\left([0, T], L^2(\Omega)\right) \cap L^p(0, T, W^{1, p}_0(\Omega)) $ to problem (3.1) satisfying (3.4). Hence we have that, by Lemma 3.3 and Lemma 2.2, there exists $ u\in C^0\left([0, T], L^2(\Omega)\right) \cap L^p(0, T, W^{1, p}_0(\Omega)) $ such that

as $ \delta \rightarrow 0^+ $. By semicontinuity, (3.4) implies (4.1)

We claim that the limit function $ u $ solves the variational inequality (1.3) in the strong form.

It is immediate to check that

Indeed, (4.4) holds since $ u_\delta(\cdot, 0) = u_0 $ a.e. in $ \Omega $ for every $ \delta > 0 $. On the other hand, if we pass to the limit as $ \delta\rightarrow 0^+ $ in (3.18) and take into account (4.2) we have $ \|(u-\psi)^-\|_{L^{ 2 \wedge p } (\Omega_T)} = 0 $ which clearly implies (4.5).

Our next goal is to prove that

as $ \delta \rightarrow 0^+ $. We test the penalized equation by $ \mathcal T_1(u_\delta-u) $ and since condition (4.5) implies

we get the following inequality

If we set $ \Phi(z): = \int_0^z \mathcal T_1(\zeta) \, \mathrm d \zeta $, by (4.4) we obtain

Because of (4.3), the latter term in the last inequality vanishes in the limit as $ \delta \rightarrow 0 $. So, as $ \Phi $ is nonnegative, we get

Again by (4.3), the right hand side of (4.7) vanishes in the limit as $ \delta \rightarrow 0 $, and so (4.7) implies

By (1.7), (3.2) and (3.3) we have

therefore, by the dominated convergence theorem and by (4.2), we get

Combining (4.8) and (4.9) and by (1.6) we get

Using again (1.6), relation (4.10) gives

and so by (4.2) we get

as $ \delta \rightarrow 0 $. By Lemma 3.1 in ^[21] we deduce that (4.6) holds.

We let $ v \in \mathcal K_{\psi}(\Omega_T) $. It is clear that $ \left[(\psi-u_\delta)^+\right]^{q-1} \mathcal T_\lambda(u_\delta-v) \leqslant 0 $ a.e. in $ \Omega_T $ and for every $ \lambda > 0 $. For this reason, if we use $ \mathcal T_\lambda(u_\delta-v) $ as a test function in (3.1) we deduce

We set $ \Phi_\lambda(z): = \int_0^z \mathcal T_\lambda(\zeta) \, \mathrm d \zeta $ and we have

We observe that Lemma 2.2 applies because of (3.4) and (3.19), so

This convergence and the Lipschitz continuity of $ \Phi_\lambda $ gives $ \Phi_\lambda(u_\delta-v)(\cdot, T) \rightharpoonup \Phi_\lambda(u-v)(\cdot, T) $ weakly in $ L^2(\Omega) $, then

On the other hand, by Fatou lemma, we are able to pass to the limit as $ \delta\rightarrow 0 $ in the third term on the left–hand side of (4.11). Indeed, for this term we know by the monotonicity condition (1.6) that the integrand is nonnegative and we have already observed that $ u_\delta $ and $ \nabla u_\delta $ converge a.e. according to (4.2) and (4.6) respectively. We only need to handle the term

This can be done arguing similarly as for the case $ \lambda = 1 $. By (1.7) we have

By (4.2) and (4.5) we obtain $ A(x, t, u_\delta \vee \psi, \nabla v)\to A(x, t, u, \nabla v) $ a.e. in $ \Omega_T $, Therefore, by the dominated convergence theorem, $ A(x, t, u_\delta \vee \psi, \nabla v)\to A(x, t, u, \nabla v) $ strongly in $ L^{p^\prime}(\Omega_T, \mathbb{R}^N) $, and this yields

Taking into account the latter relation and also (4.12) and (4.13), we can now pass to the limit as $ \delta\rightarrow 0 $ in (4.11) and obtain

Since

and also observing that

we conclude that (1.3) holds.

Next result shows that a Lewy–Stampacchia inequality can be derived under some suitable assuption, that we are going to remove later.

Proposition 4.2. Let (1.2), (1.4)–(1.16), (3.2) and (3.17) be in charge. If we also assume that

the solution $ u $ of the obstacle problem constructed in Proposition 4.1 satisfies the Lewy–Stampacchia inequality (1.17).

Proof. We define

For $ k \geqslant 1 $ we also define

Thanks to Lemma 4.3 in ^[15] we are able to test (3.1) by $ \Psi_k(x, s, u_\delta-\psi) \chi_{(0, t)} $ for $ t\in (0, T) $, obtaining

By (1.14) we have

We also have

So, taking into account (4.14), we have

We remark that

Since we have $ \left\{ g^- - \frac 1 \delta \eta_k((u_\delta-\psi)^-) < 0\right\} \subset \{ u_\delta < \psi \} $ then

By (1.6) it follows that

Hence, we deduce from (4.15)

Now, we pass to the limit as $ k\rightarrow \infty $. In particular, by using the monotone convergence theorem, we have

and also

We gather the previous relations, and (since $ t\in (0, T) $ is arbitrary) we get

Since it is clear that

we obtain

Observing that (4.2), (4.5) and (4.6) hold, then

as $ \delta \rightarrow 0 $. By (1.7), (3.2) and (3.4), $ F_\delta $ is also bounded in $ L^{p^\prime}(\Omega_T) $, hence $ F_\delta\rightharpoonup 0 $ in $ L^{p^\prime}(\Omega_T) $. We deduce

By (4.16) we obtain

Hence we have

and so

Similarly, rewriting (3.1) as follows

then

and the proof is completed.

Next result provides the one of Theorem 1.1 under the assumption (3.2) but removing condition (3.17).

Proposition 4.3. Let (1.2), (1.4)–(1.16) and (3.2) be in charge. There exists at least solution $ u \in \mathcal K_{\psi}(\Omega_T) $ to the variational inequality (1.3) satisfying $ u(\cdot, 0) = u_0 $ in $ \Omega $, the estimate (4.1) and the Lewy–Stampacchia inequality (1.17).

Proof. We know that

where $ g^\pm $ are nonnegative elements of $ L^{p'}(0, T, W^{-1, p'}(\Omega)). $ By using a regularization procedure, due to ^[7] Lemma p. 593, and Lemma 4.1 in ^[15], we find a sequence $ \{g_n^-\}_{n\in \mathbb N} $ of nonnegative functions such that

and

We define

It is clear that

as $ n\to \infty $. Due to the regularity assumptions on $ g_n^- $, we get the existence of $ u_n \in \mathcal K_{\psi}(\Omega_T) $ with $ u_n(\cdot, 0) = u_0 $ in $ \Omega $ such that for every $ v \in \mathcal K_{\psi}(\Omega_T) $ we have

Moreover, the subsequent estimate holds

and the following Lewy-Stampacchia inequality holds

Since the sequence $ \{f_n\}_{n \in \mathbb N} $ is strongly converging (and hence bounded) in $ L^{p'}(0, T, W^{-1, p'}(\Omega)) $, we obtain

for some positive constant $ C $ independent of $ n $. Moreover, the Lewy–Stampacchia inequality (4.18) implies a uniform bound of this kind

again for some positive constant $ C $ independent of $ n $. Therefore, there exists $ u\in C^0\left([0, T], L^2(\Omega)\right) \cap L^p(0, T, W^{1, p}_0(\Omega)) $ with $ u(\cdot, 0) = u_0 $ in $ \Omega $ such that

as $ n\to \infty $. Obviously (4.19) implies $ u \geqslant \psi $ a.e. in $ \Omega_T $. If we summarize, we have $ u \in \mathcal K_{\psi}(\Omega_T) $ and then $ v_n: = u_n-\mathcal T_1(u_n-u) \in \mathcal K_{\psi}(\Omega_T). $ Hence, we use $ v_n $ as a test function in (4.17) and, arguing as in the proof of Proposition 4.1, we obtain

as $ n\to \infty $. For fixed $ \lambda > 0 $ and $ v \in \mathcal K_{\psi}(\Omega_T) $ we also have $ v_{n, \lambda}: = u_n-\mathcal T_\lambda(u_n-v) \in \mathcal K_{\psi}(\Omega_T). $ Arguing again as in the proof of Proposition 4.1, we get (1.3) passing to the limit (first as $ n \rightarrow \infty $ and then as $ \lambda\rightarrow \infty $) in the inequality obtained by testing (4.17) by $ v_{n, \lambda} $.

Finally, we remove condition (3.2), i.e., we are able to prove Theorem 1.1.

Proof of Theorem 1.1. The convex set $ \mathcal K_{\psi}(\Omega_T) $ is nonempty and one can find $ w \in \mathcal K_{\psi}(\Omega_T) $ such that $ w(\cdot, 0) = \psi(\cdot, 0) $ in $ \Omega $ (see for details Remark 2.1 in ^[15]). Let us define

Hence $ \hat f \in L^{p'}(0, T, W^{-1, p'}(\Omega)) $ and $ \hat \psi $ and $ \psi $ share the same trace on $ \partial \Omega \times (0, T) $. Therefore, one can conclude

Moreover, the vector field $ \hat A $ enjoys similar properties as $ A $. This is trivial for conditions (1.6) and (1.7). As in ^[12], properties of $ A $ and Young inequality, we have for $ \varepsilon > 0 $

with a suitable $ H_1\in L^1(\Omega_T) $. Moreover, as an elementary consequence of the convexity of $ |\; |^p $, for $ 0 < \vartheta < 1 $ we find a constant $ C = C(\vartheta, p) > 0 $ such that

Hence, we get coercivity condition for $ \hat A $:

where we set

and denoted by $ \hat H $ a suitable nonnegative function in $ L^1(\Omega_T) $. Obviously, we can make $ \hat\alpha $ arbitrarily close to $ \alpha $, by choosing $ \varepsilon $ close to $ 0 $ and $ \vartheta $ close to $ 1 $. Using inequality (2.5) for $ b $ and $ \tilde b $ in place of $ f $ and $ g $, respectively, we can easily show that also $ \mathscr D_{\hat b} $ is arbitrarily close to $ \mathscr D_{ b} $, again by choosing $ \varepsilon $ close to $ 0 $ and $ \vartheta $ close to $ 1 $. Indeed, we have

By (1.16) we can also have

We observe that

We can apply Proposition 4.3 for the operator $ \hat A $. Therefore, we obtain the existence of a function $ \hat u \in \mathcal K_{\hat \psi} (\Omega_T) $ such that

and the following parabolic variational inequality

holds true for every admissible function $ \hat v \in \mathcal K_{\hat \psi} (\Omega_T) $. Since any $ v \in \mathcal K_{ \psi} (\Omega_T) $ can be rewritten as $ v = \hat v +w $ for some $ \hat v \in \mathcal K_{\hat \psi} (\Omega_T) $, by (4.20), by the definitions of $ \hat A $, $ \hat f $ and $ \hat \psi $, we see that the variational inequality (1.3) holds true with $ u: = \hat u +w $ and for any admissible function $ v \in \mathcal K_{\psi}(\Omega_T) $. The fact that $ u \in \mathcal K_{\psi}(\Omega_T) $ and $ u(\cdot, 0) = u_0 $ in $ \Omega $ is obvious, and this concludes the proof.

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). F. Farroni also acknowledges support by project Starplus 2020 Unina Linea 1 "New challenges in the variational modeling of continuum mechanics'' from the University of Naples Federico II and Compagnia di San Paolo. G. Zecca also acknowledges support by Progetto FRA 2022 "Groundwork and OptimizAtion Problems in Transport'' from the University of Naples Federico II.

Conflict of interest

The authors declare no conflict of interest.