Current and future biocompatibility aspects of biomaterials for hip prosthesis

Dedicated to Giuseppe Mingione, on the occasion of his 50th birthday.

1.   Introduction

For localized problems, many papers showed that the weak solution of elliptic and parabolic equations can be obtained with a limit of approximations by regularizing the nonlinearities, see for instance ^{[1,2,4,28,29,32]}. However, as far as we are concerned, it was hard to find a suitable reference for global problems which considered approximations on domains. In this paper, we will show that the weak solution can be obtained with a limit of approximations by regularizing the nonlinearities and approximating the domains for Dirichlet boundary value problems. Also we refer to ^[19,20] which used regularization on the nonlinearities and approximation on the convex domains for a class of nonlinear elliptic systems.

For the interested readers, we briefly explain about the mentioned papers in the previous paragraph, which are mainly related to the regularity of elliptic and parabolic problems. Acerbi and Fusco ^[1] obtained local $ C^{1, \gamma} $ for local minimizers of $ p $–energy density, where we refer to ^[35,52,53] for fundamental papers and ^[27] for generalized elliptic systems. Acerbi and Mingione ^[2] obtained local $ C^{1, \gamma} $ regularity for local minimizers with variable exponents, where we refer to ^[54] for fundamental paper and ^[3,8,16] for Calderón-Zygmund type estimates. Esposito, Leonetti and Mingione ^[32,33] obtained higher integrability results for elliptic equations with $ p $–$ q $ growth conditions, where we refer to ^[10,18,24] for the related results and ^[46,47] for Lipschitz regularity. Also we refer to ^{[9,21,22,23,25]} for double phase problems and ^[37] for a unified approach of $ p $–$ q $, Orlicz, $ p(x) $ and double phase growth conditions. Acerbi and Mingione ^[4] obtained Calderón-Zygmund type estimate for a class of parabolic systems, and we refer to ^[11,15,17] for the global results and ^[6] for Lorentz space type estimate. Duzaar and Mingione ^[28] obtained local Lipschitz regularity for nonlinear elliptic equations and a class of elliptic systems. Also Cianchi and Maz'ya ^[19,20] obtained Lipschitz regularity for a class of elliptic systems in convex domains. Duzaar and Mingione ^[29] obtained Wolff potential type estimate for nonlinear elliptic equations, and we refer to ^{[39,40,41,42,43,44,49]} for further references and ^[7] for nonlinear elliptic equations with general growth. We remark that one of the authors obtained ^[14] based on the techniques of ^[29,48].

1.1.   Parabolic equations

Suppose that $ a : \mathbb{R}^n \times \mathbb{R}^{n+1} \rightarrow \mathbb{R}^n $ satisfies

and the following ellipticity and growth conditions:

for every $ (x, t) \in \mathbb{R}^{n+1} $, for every $ \xi, \zeta \in \mathbb{R}^n $ and for some constants $ 0 < \lambda \leq \Lambda $ and $ s \geq 0 $.

To regularize the nonlinearity $ a $, we define $ \phi \in C_{c}^{\infty}(\mathbb{R}^{n}) $ as a standard mollifier:

where $ c_{1} > 0 $ is a constant chosen so that

Under the assumptions (1.1) and (1.2), let $ a_\epsilon(\xi, x, t) $ be a regularization of $ a(\xi, x, t) $:

Then $ a_\epsilon(\xi, x, t) $ satisfies the ellipticity and growth conditions and it is smooth enough, precisely,

and

for $ s_{\epsilon} = (s^{2} + \epsilon^{2})^\frac{1}{2} > 0 $. Here, the constants $ c $ are depending only on $ n $ and $ p $. It will be proved in Lemma 2.13.

As usual, we denote $ p' $ as the Hölder conjugate of $ p $ and by $ p^* $ the Sobolev exponent of $ p $. (Note that $ p^* $ can be any real number bigger than $ 1 $, provided that $ p \ge n $.) We denote $ d_{H}(X, Y) $ as the Hausdorff distance between two nonempty sets $ X $ and $ Y $, namely,

Remark 1.1. As mentioned before, $ a_{k}(\xi, x, t) $ is smooth with respect to $ \xi $ and $ x $ by Lemma 2.13. For Neumann boundary value problems, we need to consider extensions to compare weak solutions defined on different domains. In this paper, we consider Dirichlet boundary value problem with $ \gamma \in W^{1, p}(\Omega) $ to obtain the main theorem without using extensions.

We will only prove the parabolic case, because the elliptic case can be done in a similar way. To consider parabolic equations, we denote $ \Omega_{\tau} = \Omega \times [0, \tau] $ and $ \mathbb{R}^{n}_{\tau} = \mathbb{R}^{n} \times [0, \tau] $ for $ \tau \in [0, T] $, where $ T > 0 $. We write $ \left\langle {\left\langle {} \right.} \right. \cdot, \cdot \left. {\left. {} \right\rangle } \right\rangle_{\Omega} = \left\langle {\left\langle {} \right.} \right. \cdot, \cdot \left. {\left. {} \right\rangle } \right\rangle_{\langle W^{-1, p'}(\Omega), W^{1, p}_{0}(\Omega) \rangle} $ as the pairing between $ W^{-1, p'}(\Omega) $ and $ W^{1, p}_{0}(\Omega) $, where $ W^{-1, p}(\Omega) $ is the dual space of $ W^{1, p}_{0}(\Omega) $. We carefully note that $ \langle \cdot, \cdot \rangle $ stands for the inner product in $ \mathbb{R}^n $ or $ \mathbb{R}^{n+1} $. We also note that for the consistency of the notation, we usually write $ W^{1, p}_{0}(\mathbb{R}^{n}) $ instead of $ W^{1, p}(\mathbb{R}^{n}) $. Here, we remark that $ W^{1, p}_{0}(\mathbb{R}^{n}) = W^{1, p}(\mathbb{R}^{n}) $. For $ \partial_{t}w $, we mean $ \partial_{t} w \in L^{p'} \big(0, T; W^{-1, p'}(\Omega) \big) $ satisfying

We consider a sequence of functions $ \{ u_{k} \}_{k = 1}^{\infty} $ defined on the corresponding sequence of domains $ \{ \Omega^{k} \}_{k = 1}^{\infty} $ in this paper. So to use convergence on $ \{ u_{k} \}_{k = 1}^{\infty} $, we consider the zero extension as in the following definition. In this paper, '$ \to $' means the strong convergence and '$ \rightharpoonup $' means the weak convergence.

Definition 1.2. For $ 1 < p < \infty $, we say $ v_{k} \in L^{p'} (\Omega_{T}^{k}) $ $ (k \in \mathbb{N}) $ converges strongly-$ \ast $ to $ v_{\infty} \in L^{p'}(\Omega_{T}^{\infty}) $, which is denoted by $ v_{k} \in L^{p'}(\Omega_{T}^{k}) \, \overset{\ast}{\to} \, v_{\infty} \in L^{p'}(\Omega_{T}^{\infty}) $, if

for any $ \eta_{k} \in L^{p} (\Omega_{T}^{k}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ satisfying

where $ \bar{\eta}_{k} $ is the zero extension of $ \eta_{k} $ from $ \Omega_{T}^{k} $ to $ \mathbb{R}^{n}_{T} $.

Remark 1.3. In Definition 1.2, if $ \Omega^{k} = \Omega^{\infty} $ for any $ k \in \mathbb{N} $, then $ v_{k} \to v_{\infty} $ in $ L^{p'} (\Omega_{T}^{\infty}) $ is equivalent to strong-$ \ast $ convergence, see Lemma 3.1.

We use a similar definition for $ W^{-1, p'} $. We remark that $ W^{1, p}_{0}(\Omega) $ is reflexive when $ 1 < p < \infty $.

Definition 1.4. For $ 1 < p < \infty $, we say that $ v_{k} \in W^{-1, p'} (\Omega^{k}) $ $ (k \in \mathbb{N}) $ converges strongly-$ \ast $ to $ v_{\infty} \in W^{-1, p'}(\Omega^{\infty}) $, which is denoted by $ v_{k} \in W^{-1, p'} (\Omega^{k}) \, \overset{\ast}{\to} \, v_{\infty} \in W^{-1, p'}(\Omega^{\infty}) $, if

for any $ \eta_{k} \in W^{1, p}_{0}(\Omega^{k}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ satisfying

where $ \bar{\eta}_{k} $ is the zero extension of $ \eta_{k} $ from $ \Omega^{k} $ to $ \mathbb{R}^{n} $.

Definition 1.5. For $ 1 < p < \infty $, we say that $ v_{k} \in L^{p'} \big(0, T; W^{-1, p'} (\Omega^{k}) \big) $ $ (k \in \mathbb{N}) $ converges strongly-$ \ast $ to $ v_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $, denoted by $ v_{k} \in L^{p'} \big(0, T; W^{-1, p'} (\Omega^{k}) \big) \, \overset{\ast}{\to} \, v_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $, if

for any $ \eta_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ satisfying

where $ \bar{\eta}_{k} \in L^{p} \big(0, T; W_{0}^{1, p}(\mathbb{R}^{n}) \big) $ is the zero extension of $ \eta_{k} $.

For $ p > \frac{2n}{n+2} $ and an open bounded domain $ \Omega \subset \mathbb{R}^{n} $ $ (n \geq 2) $, assume that

and

Let $ u \in C \big([0, T]; L^{2}(\Omega) \big) \cap L^{p} \big(0, T; W^{1, p}(\Omega) \big) $ be the weak solution of

Here, we say that $ u \in \gamma + L^{p} \big(0, T; W_{0}^{1, p}(\Omega) \big) \cap C^{0} \big([0, T]; L^{2} (\Omega) \big) $ is the weak solution of (1.6), if

holds for any $ \varphi \in C_{0}^{\infty}(\Omega_{T}) $. Also for the initial condition, it means that

which is equivalent to $ u(x, 0) = \gamma(x, 0) $ when $ u \in C \big([0, T]; L^{2}(\Omega) \big) $.

Now, we introduce the main result in this paper.

Theorem 1.6. Let $ \Omega^{k} \subset \mathbb{R}^{n} $ $ (k \in \mathbb{N}) $ be a sequence of open bounded domains with

For $ k \in \mathbb{N} $, assume that $ \epsilon_{k} > 0 $, $ F_{k} \in L^{p}(\Omega_{T}^{k}, \mathbb{R}^{n}) $, $ f_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) $ and

satisfy that $ \lim_{k \to \infty} \epsilon_{k} = 0 $,

and

Then for the weak solution $ u_{k} \in C \big([0, T]; L^{2}(\Omega^{k}) \big) \cap L^{p} \big(0, T; W^{1, p}(\Omega^{k}) \big) $ of

where $ a_{k}(\xi, x, t) = a_{\epsilon_{k}}(\xi, x, t) $, we have that

where $ u $ is the weak solution of (1.6).

We refer to ^[13] for Calderón-Zygmund type estimates for a class of elliptic and parabolic systems with nonzero boundary data in rough domains such as Reifenberg flat domains.

Remark 1.7. For the sake of convenience and simplicity, we employ the letters $ c > 0 $ throughout this paper to denote any constants which can be explicitly computed in terms of known quantities such as $ n, p, \lambda, \Lambda $ and the diameter of the domains. Thus the exact value denoted by $ c $ may change from line to line in a given computation.

Remark 1.8. We usually denote $ \bar{g} $ as the natural zero extension of $ g $ for such space as $ L^{p}(\Omega_{T}) $ and $ L^{p'} \big(0, T; W^{-1, p'}(\Omega) \big) $ which depends on the situations.

1.2.   Elliptic equations

We also have a result for elliptic equations which corresponds to Theorem 1.6. The proof is similar to that of Theorem 1.6, and we will only state the result.

Suppose that $ a : \mathbb{R}^n \times \mathbb{R}^{n} \rightarrow \mathbb{R}^n $ satisfies

and the following ellipticity and growth conditions:

for every $ x, \xi, \zeta \in \mathbb{R}^n $ and for some constants $ 0 < \lambda \leq \Lambda $ and $ s \geq 0 $.

Under the assumptions (1.12) and (1.13), let $ a_\epsilon(\xi, x) $ be a regularization of $ a(\xi, x) $:

Then $ a_\epsilon(\xi, x) $ satisfies the ellipticity and growth conditions, such as (1.2), and it is smooth enough, precisely,

We have the following approximation results for elliptic problems.

Theorem 1.9. For $ 1 < p < \infty $ and an open bounded domain $ \Omega \subset \mathbb{R}^{n} $ $ (n \geq 2) $, assume that $ F \in L^{p}(\Omega, \mathbb{R}^{n}) $, $ f \in L^{(p^{*})'}(\Omega) $ and $ \gamma \in W^{1, p}(\Omega) $. Let $ u \in \gamma + W_{0}^{1, p}(\Omega) $ be the weak solution of

Let $ \Omega^{k} \subset \mathbb{R}^{n} $ $ (k \in \mathbb{N}) $ be a sequence of open bounded domains with

For $ k \in \mathbb{N} $, assume that $ \epsilon_{k} > 0 $, $ F_{k} \in L^{p}(\Omega^{k}, \mathbb{R}^{n}) $, $ f_{k} \in L^{(p^{*})'}(\Omega^{k}) $ and $ \gamma \in W^{1, p}(\Omega^{k}) $ satisfy that

and

Then for the weak solution $ u_{k} \in \gamma_{k} + W_{0}^{1, p}(\Omega^{k}) $ of

where $ a_{k}(\xi, x) = a_{\epsilon_{k}}(\xi, x) $, we have that

2.   Preliminaries

2.1.   Basic results about functional analysis

We use the following results related to weak convergence and weak* convergence.

Proposition 2.1. [12, Proposition 3.13 (iii)] Let $ \{ f_{i} \} $ be a sequence in $ E^{*} $. If $ f_{i} \overset{\ast}{\rightharpoonup} f $ in $ \sigma(E^{*}, E) $ then $ \{ \| f_{i} \| \} $ is bounded and $ \| f \| \leq \liminf \| f_{i} \| $.

Proposition 2.2. [12, Theorem 3.16 (Banach-Alaoglu-Bourbaki)] The closed unit ball $ B_{E^{*}} = \{ f \in E^{*} : \| f \| \leq 1 \} $ is compact in the weak-$ \ast $ topology $ \sigma (E^{*}, E) $.

One can easily check that compactness in Proposition 2.2 implies sequential compactness for metric spaces.

Proposition 2.3. If $ E^{*} $ is a metric space then any bounded sequence $ \{ f_{i} \} $ in $ E^{*} $ has a weakly-$ \ast $ convergent subsequence.

To apply Proposition 2.1 and Proposition 2.3 to Sobolev space, we use Proposition 2.4.

Proposition 2.4. [12, Proposition 8.1] $ W^{1, p} $ is a Banach space for $ 1 \leq p \leq \infty $. $ W^{1, p} $ is reflexive for $ 1 < p < \infty $ and separable for $ 1 \leq p < \infty $.

To handle the dual space of $ W^{1, p}_{0}(\Omega) $, we use [45, Corollary 10.49].

Proposition 2.5. [45, Corollary 10.49] Let $ \Omega \subset \mathbb{R}^{n} $ be an open set and $ 1 \leq p < \infty $. Then $ h \in W^{-1, p'}(\Omega) $ can be identified as

with

for some $ H = (H_{0}, H_{1}, \cdots, H_{n}) \in L^{p'}(\Omega, \mathbb{R}^{n+1}) $.

We have the following result from [51, Proposition Ⅲ.1.2], [30, Lemma 2.1] and [50, Lemma 3.1].

Proposition 2.6. [51, Proposition III.1.2] Let $ \Omega \subset \mathbb{R}^{n} $ be a bounded domain, $ t_{1} < t_{2} $ and $ p > \frac{2n}{n+2} $. Assume that $ v \in L^{p} \big(t_{1}, t_{2}; W^{1, p}_{0}(\Omega) \big) $ has a distributional derivative $ \partial_{t} v \in L^{p'} \big(t_{1}, t_{2}; W^{-1, p'}(\Omega) \big) $. Then there holds $ v \in C \big([t_{1}, t_{2}]; L^{2}(\Omega) \big) $ and moreover, the mapping $ t \mapsto \| v (\cdot, t) \|_{L^{2}(\Omega)}^{2} $ is absolutely continuous on $ [t_{1}, t_{2}] $ with

where $ \left\langle {\left\langle {} \right.} \right. \cdot, \cdot \left. {\left. {} \right\rangle } \right\rangle_{\Omega} $ denotes the dual pairing between $ W^{-1, p'}(\Omega) $ and $ W^{1, p}_{0}(\Omega) $.

2.2.   Basic inequalities on elliptic condition

We use the following basic inequality in this paper.

Lemma 2.7. [38, Lemma 3.2] For any $ q > 1 $ and $ s \geq 0 $, there exists $ \kappa_{1} = \kappa_{1}(n, q) \in (0, 1] $ such that

for any $ \kappa \in (0, \kappa_{1}] $.

We would like to emphasis that the inequalities in Lemmas 2.8 and 2.9 are obtained for $ s \geq 0 $ even when $ 1 < q < 2 $. We remark that a different proof for $ 1 < q < 2 $ was shown in [1, Lemma 2.1].

Lemma 2.8. For any $ q > 1 $ and $ s \geq 0 $, we have that

for any $ \xi, \zeta \in \mathbb{R}^{n} \setminus \{ 0 \} $, where $ c $ depends only on $ q $.

Proof. By changing variable, one can easily check that

and without loss of generality, we may assume $ |\xi| \geq |\zeta| $.

If $ q \geq 2 $, then the lemma follows from the fact that

So it only remains to prove the lemma when $ 1 < q < 2 $.

Next, suppose that $ 1 < q < 2 $. We show the lemma by considering three cases:

(1). If $ 2|\zeta - \xi| \leq |\xi| $, then for any $ \tau \in [0, 1] $ we have

because we assumed that $ |\xi| \geq |\zeta| $, which implies

and the lemma is proved for the first case.

(2). If $ |\xi| \leq 2|\zeta - \xi| \leq 2s $, then we obtain

which implies

and the lemma is proved for the second case.

(3). Suppose that $ |\xi| \leq 2 |\zeta - \xi| $ and $ s < |\zeta - \xi| $. One can easily check that

which implies

Then by changing variables, we obtain

where

By changing variables, we discover that

Similarly, we have

Since $ |\zeta| \leq |\xi| \leq 2 |\zeta - \xi| $ and $ s < |\zeta - \xi| $, we have $ |\xi|^{2} + |\zeta|^{2} + s^{2} \leq 9 |\zeta - \xi|^{2} $, and

By the above three inequalities and (2.1), we find that the lemma holds when $ |\xi| \leq 2 |\zeta-\xi| \text{ and } s < |\zeta - \xi| $. This completes the proof.

Lemma 2.9. For any $ q > 1 $ and $ s \geq 0 $, we have that

for any $ \xi, \zeta \in \mathbb{R}^{n} \setminus \{ 0 \} $, where $ c $ depends only on $ q $.

Proof. One can easily check that

If $ 1 < q < 2 $, then

which prove the lemma for $ 1 < q < 2 $.

To prove the lemma for the case $ q \geq 2 $, we assume that $ |\xi| \geq |\zeta| $ without loss of generality. Then for $ \tau \in [0, 1/4] $, we have

So we obtain

which prove the lemma for $ q \geq 2 $. This completes the proof.

To compare $ a(\xi, x, t) $ and $ a(\zeta, x, t) $, we use the following lemma.

Lemma 2.10. Under the assumptions (1.1) and (1.2), we have

for any $ \xi, \zeta \in \mathbb{R}^{n} $.

Proof. We fix any $ \xi, \zeta \in \mathbb{R}^{n} $. If $ |\xi| = 0 $ or $ |\zeta| = 0 $ then the lemma holds trivially from (1.1) and (1.2). So we assume that $ \xi, \zeta \in \mathbb{R}^{n} \setminus \{ 0 \} $. Since $ |\xi - \zeta|^\frac{1}{p-1} \leq c (|\xi|^{2} + |\zeta|^{2} + s^{2})^\frac{1}{2(p-1)} $, we have from (1.2) and Lemma 2.8 that

Since $ \xi, \zeta \in \mathbb{R}^{n} $ were arbitrary chosen, the lemma follows.

We show the following well-known inequality. We remark that a different proof for $ 0 < q < 2 $ was shown in [1, Lemma 2.1] and [36, Lemma 2.1].

Lemma 2.11. For any $ q > 0 $ and $ s \geq 0 $, we have that

and

for any $ \xi, \zeta \in \mathbb{R}^{n} $, where $ c $ depends only on $ q $.

Proof. We fix any $ \xi, \zeta \in \mathbb{R}^{n} $. If $ |\xi| = 0 $ or $ |\zeta| = 0 $ then the lemma holds trivially. So we assume that $ \xi, \zeta \in \mathbb{R}^{n} \setminus \{ 0 \} $. Then

By taking $ \frac{q}{2} + 1 \in (1, \infty) $ instead for $ q \in (1, \infty) $ in Lemma 2.8,

Also we get

If $ 0 < q \leq 2 $ then $ 1 = \frac{ 2-q}{2} + \frac{q}{2} $ and $ \frac{2-q}{2} \geq0 $. Also if $ q > 2 $ then $ \frac{q-2}{2} \geq 0 $. Thus

By taking $ \frac{q}{2} + 1 \in (1, \infty) $ instead for $ q \in (1, \infty) $ in Lemma 2.9,

Since $ \xi, \zeta \in \mathbb{R}^{n} $ were arbitrary chosen, the lemma follows.

We will use the following lemma.

Lemma 2.12. For any $ q > 1 $ and $ s \geq 0 $, we have that

for any $ \xi, \zeta \in \mathbb{R}^{n} $, where $ c $ only depends on $ q $.

Proof. Fix any $ \xi, \zeta \in \mathbb{R}^{n} $. By taking $ 2q-2 > 0 $ instead of $ q \left(>0 \right) $ in Lemma 2.11,

By that $ |\xi - \zeta|^\frac{1}{q-1} \leq c \left(|\xi|^{2} + |\zeta|^{2} + s^{2} \right)^\frac{1}{2(q-1)} $,

Since $ \xi, \zeta \in \mathbb{R}^{n} $ were arbitrary chosen, the lemma follows.

2.3.   Regularization on the nonlinearities

To find the ellipticity and growth conditions of $ a_{\epsilon} (\xi, x, t) $ in (1.5), we follow the approach in the proof of [31, Lemma 2] and [32, Lemma 3.1].

Lemma 2.13. For (1.5), we have

and

for $ s_{\epsilon} = (s^{2} + \epsilon^{2})^\frac{1}{2} $. Here, the constants $ c $ are depending only on $ n $ and $ p $.

Proof. Fix a vector $ \xi \in \mathbb{R}^{n} $. Since $ a(\xi, x, t) $ is $ C^{1} $-regular in $ \xi \in \mathbb{R}^{n} $ for every $ x \in \mathbb{R}^{n} $, we find that $ a_{\epsilon}(\xi, x, t) $ is $ C^{1} $-regular in $ \xi \in \mathbb{R}^{n} $ for every $ x \in \mathbb{R}^{n} $. Also by changing variable, we have from (1.5) that

which implies

Moreover, from (1.2), the fact that $ \mathrm{supp \, } \phi \subset \overline{B_{1}} $ and

for any $ m \geq 0 $, which implies that

for any $ m \geq 0 $. Similarly, by changing variable, we have from (1.5) that

and one can obtain that

So $ a_{\epsilon}(\xi, x, t) $ is $ C^{\infty} $-regular in $ \xi \in \mathbb{R}^{n} $ for every $ (x, t) \in \mathbb{R}^n $ and $ a_{\epsilon}(\xi, x, t) $ is $ C^{\infty} $-regular in $ x \in \mathbb{R}^{n} $ for every $ \xi \in \mathbb{R}^n $ and $ t \in \mathbb{R} $. Also the second inequality in (2.3) follows.

From (1.2), (1.5) and the fact that $ \mathrm{supp \, } \phi \subset \overline{B_{1}} $, we have

and the third inequality in (2.3) holds.

It only remains to prove the first inequality in (2.3). In view of (1.5), we have

If $ 16 \epsilon^{2} \geq |\xi|^{2} + s^{2} $, then by changing variables and (1.5), we obtain

and from the fact that $ 16 \epsilon^{2} \geq |\xi|^{2} + s^{2} $, we have $ 17 \epsilon^{2} \geq |\xi|^{2} + \epsilon^{2} + s^{2} $ and

So we discover that the first inequality in (2.3) holds for the case $ 16 \epsilon^{2} \geq |\xi|^{2} + s^{2} $.

On the other-hand, if $ 16 \epsilon^{2} \leq |\xi|^{2} + s^{2} $, then we have

and $ \mathrm{supp \, } \phi \subset \overline{B_{1}} $ implies

which implies that

We claim that if $ 16 \epsilon^{2} \leq |\xi|^{2} + s^{2} \text{ and } |y| \leq 1 $ then

If $ p \geq 2 $, then the claim (2.7) holds trivially. If $ 1 < p < 2 $, then $ 16 \epsilon^{2} \leq |\xi|^{2} + s^{2} $ implies

and we find that the claim (2.7) holds. Thus the claim (2.7) is proved. In light of (2.6) and (2.7), we have that if $ 16 \epsilon^{2} \leq |\xi|^{2} + s^{2} $ then

Thus the first inequality in (2.3) follows from (2.4), (2.5) and (2.8). This completes the proof.

Later, we will apply the gradient of the weak solution in Lemma 2.14 by considering a zero extension from $ \Omega_{T} $ to $ \mathbb{R}^{n}_{T} $.

Lemma 2.14. For any $ H \in L^{p}(\Omega_{T}, \mathbb{R}^{n}) $, we have that

Proof. Fix $ \delta > 0 $. From (1.5), we have

Let $ \tilde{\Omega}_{\epsilon} = \{ x \in \Omega : \mathrm{dist} \left(x, \partial \Omega \right) > \epsilon \} $ and $ \tilde{\Omega}_{\epsilon, T} = \tilde{\Omega}_{\epsilon} \times [0, T] $. Since $ H \in L^{p}(\Omega_{T}, \mathbb{R}^{n}) $, there exists $ \epsilon_{\delta} > 0 $ such that if $ \epsilon \in (0, \epsilon_{\delta}] $ then

which implies that

for any $ \epsilon \in (0, \epsilon_{\delta}] $. Thus

Since $ \delta > 0 $ was arbitrary chosen, we get

We now estimate $ a(H, \cdot) - a_{\epsilon}(H, \cdot) $ on $ \tilde{\Omega}_{\epsilon, T} $. By the triangle inequality,

where

We want to prove that the left-hand side of (2.10) goes to the zero as $ \epsilon \searrow 0 $.

To handle $ I $, we use the standard approximation by mollifiers, see for instance [34, C. Theorem 6], to find that

where we used that $ a(H, \cdot) \in L^{\frac{p}{p-1}}(\Omega_{T}) $ and $ \int_{\mathbb{R}^n} \phi(y) \, dy = 1 $, which implies that

To handle $ II $, we apply Hölder's inequality and Lemma 2.10 to find that

We apply Hölder's inequality to find that

and by using that $ H \in L^{p}(\Omega_{T}, \mathbb{R}^{n}) $, we obtain that

which implies that

Last, to handle $ III $, we find from Lemma 2.10 that

where we used that $ \mathrm{supp} \, \phi \subset \overline{B_{1}} $ from (1.3). So by that $ \int_{\mathbb{R}^n} \phi(y) \, dy = 1 $,

So we again use Hölder's inequality to find that

By using $ H \in L^{p}(\Omega_{T}, \mathbb{R}^{n}) $, we obtain that

which implies that

By combining (2.10), (2.11), (2.12) and (2.13), we find from that

and the lemma holds from (2.9).

3.   Regularization of nonlinear parabolic equations

This section is devoted to the proof of our main result, Theorem 1.6. We start with proving our main tools for convergence lemmas for the zero extensions, Lemmas 3.1–3.7. Then we apply these tools to obtain the convergence lemmas, Lemmas 3.8–3.10. To conclude our main result, we apply an indirect method. By negating the conclusion of Theorem 1.6, we show that (3.1) contradicts Lemma 3.9 and Lemma 3.10.

Let $ \bar{u}_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) \cap L^{\infty} \big(0, T; L^{2}(\mathbb{R}^{n}) \big) $ be the zero extension of $ u_{k} - \gamma_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) \cap L^{\infty} \big(0, T; L^{2}(\Omega^{k}) \big) $ in Theorem 1.6. Also we define $ \bar{u} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) \cap L^{\infty} \big(0, T; L^{2}(\mathbb{R}^{n}) \big) $ as the zero extension of $ u - \gamma \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega) \big) \cap L^{\infty} \big(0, T; L^{2}(\Omega) \big) $ in (1.6). To prove Theorem 1.6, we will assume that the conclusion of Theorem 1.6 does not hold. Then there exist $ \delta_{0} > 0 $ and a subsequence, which will be still denoted as $ u_{k} $ $ (k \in \mathbb{N}) $, such that

So by (1.7) and (1.9), it follows that

Later, we will show that a contradiction occurs due to (3.1).

To prove Theorem 1.6, we first derive the energy estimates for regularized parabolic problems in (1.10). We test (1.10) by $ u_{k}- \gamma_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) \cap C \big([0, T]; L^{2} (\Omega^{k}) \big) $ to find that

for any $ \tau \in [0, T] $, which implies that

for any $ \tau \in [0, T] $. So by Poincaré's inequality and Lemma 2.7,

Here, the constant $ c > 0 $ for Poincaré's inequality only depends on the size of the domain and $ 1 < p < \infty $, see [5, Theorem 6.30]. By taking $ \bar{u}_{k} = u_{k} - \gamma_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) \cap L^{\infty} \big(0, T; L^{2}(\Omega^{k}) \big) $,

The domain $ \Omega^{k} $ depends on the function $ \bar{u}_{k} $ $ (k \in \mathbb{N}) $. To deal with the convergence of the functions, we need to consider the domain of the functions. It is the main reason why we adapted Definitions 1.2–1.5.

To use the compactness method, we need to show that the right-hand side of (3.2) is bounded uniformly. To do it, we use the zero extensions to $ \mathbb{R}^{n}_{T} $, which makes the domain of the functions independent of $ k \in \mathbb{N} $.

Let $ \bar{v}_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ be the zero extensions of $ v_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) $ from $ \Omega_{T}^{k} $ to $ \mathbb{R}^{n}_{T} $. Also for $ h_{k} \in W^{-1, p'}(\Omega^{k}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $, we define $ \bar{h}_{k} \in W^{-1, p'}(\mathbb{R}^{n}) $ which corresponds to the zero extension in Corollary 3.3. Under the assumption (1.7), we obtain the following results.

$ (1) $ [Lemma 3.1] If $ v_{k} \in L^{q}(\Omega_{T}^{k}) \ \overset{\ast}{\to} \ v_{\infty} \in L^{q}(\Omega_{T}^{\infty}) $ $ (1 < q < \infty) $ then

$ (2) $ [Lemma 3.4] If $ h_{k} \in W^{-1, p'}(\Omega^{k}) \ \overset{\ast}{\to} \ h_{\infty} \in W^{-1, p'}(\Omega^{\infty}) $ then

$ (3) $ [Lemma 3.5] If $ h_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) \ \overset{\ast}{\to} \ h_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $ then

$ (4) $ [Lemma 3.6] If the sequence $ \| v_{k} \|_{L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) } $ $ (k \in \mathbb{N}) $ is bounded then there exists $ v_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $ with

$ (5) $ [Lemma 3.7] If the sequence $ \| v_{k} \|_{ L^{\infty} \big(0, T; L^{2}(\Omega^{k}) \big) } $ $ (k \in \mathbb{N}) $ is bounded then there exists $ v_{\infty} \in L^{\infty} \big(0, T; L^{2}(\Omega^{\infty}) \big) $ with

We apply Lemmas 3.1–3.7 to (3.2) as follows. By using Lemma 3.1, we will show that the zero extensions of $ |F_{k}|^{p-2}F_{k} $, $ \gamma_{k} $ and $ D\gamma_{k} $ converge strongly-$ \ast $. By using Lemma 3.5, we will show that the zero extensions of $ f_{k} $ and $ \partial_{t} \gamma_{k} $ converge strongly-$ \ast $. With Lemma 3.6, the existence of weakly-$ \ast $ converging subsequence of $ \partial_{t} \bar{u}_{k} $ in $ L^{p'} \big(0, T; W^{-1, p'}(\mathbb{R}^{n}) \big) $ will be obtained. Also with Lemma 3.7, the existence of weakly-$ \ast $ converging subsequence of $ \bar{u}_{k} $ in $ L^{\infty} \big(0, T; L^{2}(\mathbb{R}^{n}) \big) $ will be obtained.

We prove our main tools for convergence lemmas. From now on, we denote $ 1_{E} $ as the indicator function on the set $ E $.

Lemma 3.1. With the assumption (1.7), suppose that $ 1 < q < \infty $ and $ N \geq 1 $. If

then

where $ \bar{V}_{k} \in L^{q'}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $ is the zero extension of $ V_{k} \in L^{q'}(\Omega^{k}_{T}, \mathbb{R}^{N}) $.

Proof. Suppose that $ V_{k} \in L^{q'}(\Omega_{T}^{k}, \mathbb{R}^{N}) \ \overset{\ast}{\to} \ V_{\infty} \in L^{q'}(\Omega_{T}^{\infty}, \mathbb{R}^{N}) $. By (1.7),

for any $ \bar{\eta} \in L^{q}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $. So by Definition 1.2, we have that

which implies that

Suppose the lemma does not hold. Then there exist $ \delta > 0 $ and a subsequence (which will be still denoted as $ \{ \bar{V}_{k} \}_{k = 1}^{\infty} $) such that

Choose $ \bar{\eta}_{k} = |\bar{V}_{k} - \bar{V}_{\infty}|^{q'-2}(\bar{V}_{k} - \bar{V}_{\infty}) $ then

Since $ (\bar{V}_{k} - \bar{V}_{\infty}) \rightharpoonup 0 $ in $ L^{q'}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $ and any weakly convergent sequence is bounded, we see that $ \{ \bar{\eta}_{k} \}_{k = 1}^{\infty} $ is bounded in $ L^{q}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $. So there exists a subsequence (which will be still denoted as $ \{ \bar{\eta}_{k} \}_{k = 1}^{\infty} $) such that

for some $ \bar{\eta}_{\infty} \in L^{q}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $. By (1.7) and that $ (\bar{V}_{k} - \bar{V}_{\infty}) \rightharpoonup 0 $ in $ L^{q'}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $,

Also we have that

because for any $ \tilde{V} \in L^{q'}(\mathbb{R}^{n}_{T}, \mathbb{R}^{N}) $,

which holds from $ |\Omega^{k} \setminus \Omega| \to 0 $ and $ |\Omega \setminus \Omega^{k}| \to 0 $ by (1.7). From (3.5) and that $ V_{k} \in L^{q'}(\Omega_{T}^{k}, \mathbb{R}^{N}) \ \overset{\ast}{\to} \ V_{\infty} \in L^{q'}(\Omega_{T}^{\infty}, \mathbb{R}^{N}) $, we use Definition 1.2 to find that

which implies that

On the other-hand, by (3.4), we find that

which contradicts (3.6). So the lemma follows.

We have the following characterization for $ h \in W^{-1, p'}(\Omega) $.

Lemma 3.2. With the assumption (1.7), suppose that $ h \in W^{-1, p'}(\Omega) $ $ (1 < p < \infty) $. Then there exists $ v \in W^{1, p}_{0}(\Omega) $ such that

for any $ \varphi \in W^{1, p}_{0}(\Omega) $. In addition, we have that $ \| h \|_{W^{-1, p'}(\Omega)} = \| v \|_{W^{1, p}_{0}(\Omega)}^{p-1} $.

Proof. Since $ h \in W^{-1, p'}(\Omega) $, there exists $ H = (H_{0}, H_{1}, \cdots, H_{n}) \in L^{p'} (\Omega, \mathbb{R}^{n+1}) $ satisfying

by Proposition 2.5. Let $ v \in W^{1, p}_{0}(\Omega) $ be the weak solution of

Then for any $ \varphi \in W^{1, p}(\Omega) $, we get

So by the definition of $ \| \cdot \|_{W^{-1, p'}(\Omega)} $,

By taking $ \varphi = \frac{ v }{ \| v \|_{W^{1, p}_{0}(\Omega)} } \in W^{1, p}_{0}(\Omega) $, we get

By combining the above two estimates, we get $ \| h \|_{W^{-1, p'}(\Omega)} = \| v \|_{W^{1, p}_{0}(\Omega)}^{p-1} $.

We extend $ h \in L^{p'} \big(0, T; W^{-1, p'}(\Omega) \big) $ to $ \bar{h} \in L^{p'} \big(0, T; W^{-1, p'}(\mathbb{R}^{n}) \big) $ in Corollary 3.3, which can be viewed as a natural zero extension because of (3.7).

Corollary 3.3. With the assumption (1.7), suppose that $ h \in W^{-1, p'}(\Omega) $ $ (1 < p < \infty) $. Then for $ v \in W^{1, p}_{0}(\Omega) $ in Lemma 3.2, one can define $ \bar{h} \in W^{-1, p'}(\mathbb{R}^{n}) $ as

for any $ \bar{\varphi} \in W^{1, p}_{0}(\mathbb{R}^{n}) $, where $ \bar{v} \in W^{1, p}_{0}(\mathbb{R}^{n}) $ is the zero extension of $ v \in W^{1, p}_{0}(\Omega) $. Moreover, we have that

for any $ \varphi \in W^{1, p}_{0}(\Omega) $ and the zero extension $ \bar{\varphi} \in W^{1, p}_{0}(\mathbb{R}^{n}) $ of $ \varphi \in W^{1, p}_{0}(\Omega) $. In addition,

In Definition 1.4, we defined a convergence for a sequence of the domains, say $ h_{k} \in W^{-1, p'}(\Omega^{k}) \overset{\ast}{\to} h_{\infty} \in W^{-1, p'}(\Omega^{\infty}) $. But this convergence implies strong convergence by considering the zero extension in Corollary 3.3 as in the next lemmas.

Lemma 3.4. Under the assumption (1.7) and $ 1 < p < \infty $, if $ h_{k} \in W^{-1, p'}(\Omega^{k}) \, \overset{\ast}{\to} \, h_{\infty} \in W^{-1, p'}(\Omega^{\infty}) $ then

and

for $ \bar{v}_{k} \in W^{1, p}_{0}(\mathbb{R}^{n}) $ and $ \bar{h}_{k} \in W^{-1, p'}(\mathbb{R}^{n}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ in Corollary 3.3.

Proof. By using Corollary 3.3, define $ \bar{h}_{k} \in W^{-1, p'}(\mathbb{R}^{n}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ as

for any $ \bar{\varphi} \in W^{1, p}_{0}(\mathbb{R}^{n}) $. Here, $ v_{k} \in W^{1, p}_{0}(\Omega^{k}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ is defined in Lemma 3.2 and $ \bar{v}_{k} \in W^{1, p}_{0}(\mathbb{R}^{n}) $ the zero extension of $ v_{k} \in W^{1, p}_{0}(\Omega^{k}) $. Moreover,

For $ k \in \mathbb{N} \cup \{ \infty \} $, let $ V_{k} = \left(|v_{k}|^{p-2} v_{k}, |Dv_{k}|^{p-2} Dv_{k} \right) \in L^{p'} (\Omega^{k}, \mathbb{R}^{n+1}) $ and $ \bar{V}_{k} \in L^{p'} (\mathbb{R}^{n}, \mathbb{R}^{n+1}) $ be the zero extension of $ V_{k} $.

Suppose that (3.9) does not hold. Then there exist $ \delta > 0 $ and a subsequence, which will be still denoted as $ \{ \bar{v}_{k} \}_{k = 1}^{\infty} $, such that

Since $ \bar{v}_{k} \left \| \bar{v}_{k} \right \|_{W^{1, p}_{0}(\mathbb{R}^{n})} ^{-1} $ is bounded in $ W^{1, p}_{0}(\mathbb{R}^{n}) $, there exists a subsequence, which will be still denoted as $ \bar{v}_{k} \left \| \bar{v}_{k} \right \|_{W^{1, p}_{0}(\mathbb{R}^{n})} ^{-1} $ $ (k \in \mathbb{N}) $, such that

for some $ v_{0} \in W^{1, p}_{0}(\Omega^{\infty}) $ and the zero extension $ \bar{v}_{0} \in W^{1, p}_{0}(\mathbb{R}^{n}) $ of $ v_{0} \in W^{1, p}_{0}(\Omega^{\infty}) $. By taking $ \bar{\varphi} = \bar{v}_{k} \left \| \bar{v}_{k} \right \|_{W^{1, p}_{0}(\mathbb{R}^{n})} ^{-1} $ in (3.10), we find from Definition 1.4 that

So $ \bar{v}_{k} $ is bounded in $ W^{1, p}_{0}(\mathbb{R}^{n}) $, and there exist $ \bar{v}_{0} \in W^{1, p}_{0}(\mathbb{R}^{n}) $, $ \bar{V}_{0} \in L^{p'}(\mathbb{R}^{n}, \mathbb{R}^{n+1}) $ and a subsequence, which will be still denoted as $ \{ \bar{v}_{k} \}_{k = 1}^{\infty} $, such that

Recall that $ \bar{V}_{k} = \left(|\bar{v}_{k}|^{p-2} \bar{v}_{k}, |D\bar{v}_{k}|^{p-2} D\bar{v}_{k} \right) \in L^{p'} (\mathbb{R}^{n}, \mathbb{R}^{n+1}) $ is the zero extension of $ V_{k} = \left(|v_{k}|^{p-2} v_{k}, |Dv_{k}|^{p-2} Dv_{k} \right) \in L^{p'} (\Omega^{k}, \mathbb{R}^{n+1}) $. Because of the assumption (1.7), one can also show that

Also by (1.7),

From (3.13), (3.14) and Definition 1.4, we obtain that

for any $ \varphi \in C_{c}^{\infty}(\Omega^{\infty}) $ and the zero extension $ \bar{\varphi} \in C_{c}^{\infty}(\mathbb{R}^{n}) $ of $ \varphi \in C_{c}^{\infty}(\Omega^{\infty}) $. Also from (3.12), (3.13) and (3.14), we obtain that

for any $ \varphi \in C_{c}^{\infty}(\Omega^{\infty}) $ and the zero extension $ \bar{\varphi} \in C_{c}^{\infty}(\mathbb{R}^{n}) $ of $ \varphi \in C_{c}^{\infty}(\Omega^{\infty}) $. Thus

for any $ \varphi \in C_{c}^{\infty}(\Omega^{\infty}) $. For any $ \varphi \in W^{1, p}_{0}(\Omega^{\infty}) $, there exists $ \varphi_{\epsilon} \in C_{c}^{\infty}(\Omega^{\infty}) $ with $ \| \varphi - \varphi_{\epsilon} \|_{W^{1, p}_{0}(\Omega^{\infty})} < \epsilon $, which implies that

Since $ \epsilon > 0 $ was arbitrary chosen, we find that

for any $ \varphi \in W^{1, p}_{0}(\Omega^{\infty}) $.

Fix $ \varphi \in C_{c}^{\infty}(\Omega^{\infty}) $. By (3.14), there exists $ K \in \mathbb{N} $ with

By a direct calculation, it follows that

for any $ k \geq K $. By (3.12) and (3.14), $ (\bar{v}_{k} - \bar{v}_{\infty} \varphi) \rightharpoonup (\bar{v}_{0} - \bar{v}_{\infty} \varphi) $ in $ W^{1, p}_{0}(\mathbb{R}^{n}) $. So by Definition 1.4,

and

which implies that

By (3.12),

By combining (3.17) and (3.18), we use (3.15) to find that

because of that $ \bar{v}_{\infty} (1-\varphi) \in W^{1, p}_{0}(\Omega^{\infty}) $. Then by Lemma 2.11,

but this contradicts (3.11) and we find that (3.9) holds. So by Lemma 2.12,

This implies that

and the lemma follows.

Lemma 3.5. Under the assumption (1.7) and $ 1 < p < \infty $, suppose that $ h_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) \ \overset{\ast}{\to} \ h_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $. Then

and

for $ \bar{v}_{k} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $ and $ \bar{h}_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\mathbb{R}^{n}) \big) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ in Corollary 3.3.

Proof. For any $ t \in [0, T] $, by using Corollary 3.3, define $ \bar{h}_{k}(\cdot, t) \in W^{-1, p'}(\mathbb{R}^{n}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ as

for any $ \bar{\varphi} (\cdot, t) \in W^{1, p}_{0}(\mathbb{R}^{n}) $. Here, $ v_{k} (\cdot, t) \in W^{1, p}_{0}(\Omega^{k}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ is defined in Lemma 3.2 and $ \bar{v}_{k} (\cdot, t) \in W^{1, p}_{0}(\mathbb{R}^{n}) $ is the zero extension of $ v_{k} (\cdot, t) \in W^{1, p}_{0}(\Omega^{k}) $.

For any $ t \in [0, T] $, let $ \bar{V}_{k}(\cdot, t) \in L^{p'} (\mathbb{R}^{n}, \mathbb{R}^{n+1}) $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ be the zero extension of

Suppose that (3.20) does not hold. Then there exist $ \delta > 0 $ and a subsequence, which will be still denoted as $ \{ \bar{v}_{k} \}_{k = 1}^{\infty} $, such that

Since $ \bar{v}_{k} \left \| \bar{v}_{k} \right \|_{L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big)} ^{-1} $ $ (k \in \mathbb{N}) $ is bounded in $ L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $, there exist $ v_{0} \in L^{p} \big(0, T; W^{1, p}_{0} (\Omega^{\infty}) \big) $ and a subsequence, which will be still denoted as $ \bar{v}_{k} \left \| \bar{v}_{k} \right \|_{L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big)} ^{-1} $ $ (k \in \mathbb{N}) $, such that

where $ \tilde{v}_{0} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $ is the zero extension of $ v_{0} \in L^{p} \big(0, T; W^{1, p}_{0} (\Omega^{\infty}) \big) $. By a direct calculation and Corollary 3.3,

Since $ v_{k}(\cdot, t) \left \| \bar{v}_{k} \right \|_{L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big)} ^{-1} \in W^{1, p}_{0}(\Omega^{k}) $ $ (k \in \mathbb{N}) $, we find from (3.8) in Corollary 3.3 and Definition 1.5 that

By taking $ \varphi = \bar{v}_{k} \left \| \bar{v}_{k} \right \|_{L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big)} ^{-1} $ in (3.21), we combine the above equality and limit to find that

So $ \bar{v}_{k} $ is bounded in $ L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $, and there exists a subsequence, which will be still denoted as $ \{ \bar{v}_{k} \}_{k = 1}^{\infty} $, such that

where $ \bar{v}_{0} \in L^{p}(\mathbb{R}^{n}_{T}) $ is weakly differentiable in $ \mathbb{R}^{n}_{T} $ with respect to $ x $-variable. Because of the assumption (1.7), one can also show that

Let $ [w]_{h}(\cdot, t) = \frac{1}{h} \int_{0}^{h} w(\cdot, t + \tau) \, d\tau $ be Steklov average of $ w $. In view of (1.7),

By (3.21) and Definition 1.5, it follows that

for any $ \varphi (\cdot, t) \in C_{c}^{\infty}(\Omega^{\infty}) $. By (3.24) and (3.26),

for any $ \varphi (\cdot, t) \in C_{c}^{\infty}(\Omega^{\infty}) $. Thus

for any $ \varphi (\cdot, t) \in C_{c}^{\infty}(\Omega^{\infty}) $. For any $ \varphi (\cdot, t) \in W^{1, p}_{0}(\Omega^{\infty}) $, there exists $ \varphi_{\epsilon} (\cdot, t) \in C_{c}^{\infty}(\Omega^{\infty}) $ with $ \| \varphi (\cdot, t) - \varphi_{\epsilon} (\cdot, t) \|_{W^{1, p}_{0}(\Omega^{\infty})} < \epsilon $. So we find that

for any $ \varphi (\cdot, t) \in W^{1, p}_{0}(\Omega^{\infty}) $ and the zero extension $ \bar{\varphi} (\cdot, t) \in W^{1, p}_{0}(\mathbb{R}^{n}) $ of $ \varphi (\cdot, t) \in W^{1, p}_{0}(\Omega^{\infty}) $. Since $ \epsilon > 0 $ was arbitrary chosen, we find from (3.25) that

for any $ \varphi (\cdot, t) \in W^{1, p}_{0}(\Omega^{\infty}) $. We now integrate it with respect to time variable $ t $ to find that

for any $ 0 < h < \epsilon < T $ and $ \varphi \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega) \big) $. Since $ V_{\infty} - V_{0} \in L^{p'}(\Omega_{T}^{\infty}) $, we use [26, Lemma 3.2] to find that

for any $ 0 < \epsilon < T $ and $ \varphi \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{\infty}) \big) $. Thus

for any $ \varphi \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{\infty}) \big) $.

Fix $ \varphi (\cdot, t) \in C_{c}^{\infty}(\Omega^{\infty}) $. By (3.26), there exists $ K \in \mathbb{N} $ with

By a direct calculation,

Also by (3.24), $ (\bar{v}_{k} - \bar{v}_{\infty} \varphi, D [\bar{v}_{k} - \bar{v}_{\infty} \varphi]) \rightharpoonup (\bar{v}_{0} - \bar{v}_{\infty} \varphi, D[\bar{v}_{0} -\bar{v}_{\infty} \varphi]) $ in $ L^{p}(\mathbb{R}^{n}_{T}) $. So by Definition 1.5,

and

which implies that

By (3.24),

By combining (3.28), (3.29) and (3.30), we use (3.27) to find that

because of that $ \bar{v}_{\infty} (1-\varphi) \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{\infty}) \big) $. So by Lemma 2.11 and (3.22),

but this contradicts (3.23) and we find that (3.20) holds. Then by Lemma 2.12

which implies that

and the lemma follows.

To obtain a weak convergence for $ \partial_{t} u_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) $ $ (k \in \mathbb{N}) $, we consider the zero extension in Corollary 3.3. We remark that

for any $ \eta \in W^{1, p}_{0}(\Omega) $ and the zero extension $ \bar{\eta} \in W^{1, p}_{0}(\mathbb{R}^{n}) $ of $ \eta \in W^{1, p}_{0}(\Omega) $, where $ \bar{h} $ is defined in Corollary 3.3.

Lemma 3.6. Under the assumption (1.7) and $ 1 < p < \infty $, let $ \Omega^{k} \subset \mathbb{R}^{n} $ $ (k \in \mathbb{N}) $ be a sequence of open bounded domains. If $ v_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) $ $ (k \in \mathbb{N}) $ satisfy

for some $ M > 0 $, then there exists $ v_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $ such that

where $ \bar{v}_{k} $ $ (k \in \mathbb{N} \cup \{ \infty \}) $ is defined in Corollary 3.3, which implies that

for any $ \bar{\eta} \in L^{p} \big(0, T; W^{1, p}_{0} (\mathbb{R}^{n}) \big) $.

Proof. Since $ v_{k} \in L^{p'} \big(0, T; W^{-1, p'}_{0}(\Omega^{k}) \big) $ $ (k \in \mathbb{N}) $, for each $ t \in [0, T] $, there exists $ V_{k}(\cdot, t) \in L^{p'} (\Omega^{k}, \mathbb{R}^{n+1}) $ such that

by Proposition 3.2. Moreover,

for any $ t \in [0, T] $. So for $ t \in [0, T] $, choose $ V_{k}(\cdot, t) \in L^{p'} (\Omega^{k}, \mathbb{R}^{n+1}) $ $ (k \in \mathbb{N}) $ so that

which implies that

for any $ k \in \mathbb{N} $.

Let $ \bar{V}_{k} $ be the zero extension of $ V_{k} $ from $ \Omega_{T}^{k} $ to $ \mathbb{R}^{n}_{T} $. Since $ \| \bar{V}_{k} \|_{ L^{p'}(\mathbb{R}^{n}_{T}, \mathbb{R}^{n+1}) } \leq 2M $ $ (k \in \mathbb{N}) $, by Proposition 2.3, there exists a weakly convergent subsequence, which will be still denoted by $ \{ \bar{V}_{k} \}_{k = 1}^{\infty} $, which converges to $ \bar{V}_{\infty} \in L^{p'}(\mathbb{R}^{n}_{T}, \mathbb{R}^{n+1}) $, say

which implies that

for any $ \bar{\eta} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $. Then one can check from (1.7) that $ \bar{V}_{\infty} = 0 $ a.e. in $ \mathbb{R}^{n}_{T} \setminus \Omega_{T}^{\infty} $. So define $ v_{\infty} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{\infty}) \big) $ as

for any $ \eta \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{\infty}) \big) $. Then by Corollary 3.3,

and

for any $ \bar{\eta} \in L^{p} \big(0, T; W^{1, p}_{0}(\mathbb{R}^{n}) \big) $. So the lemma follows from (3.33).

Lemma 3.7. Under the assumption (1.7) and $ 1 < p < \infty $, let $ \Omega^{k} \subset \mathbb{R}^{n} $ $ (k \in \mathbb{N}) $ be a sequence of open bounded domains. If $ v_{k} \in L^{\infty} \big(0, T; L^{2}(\Omega^{k}) \big) $ $ (k \in \mathbb{N}) $ satisfy

for some $ M > 0 $, then there exists $ v_{\infty} \in L^{\infty} \big(0, T; L^{2}(\Omega^{\infty}) \big) $ such that

where $ \bar{v}_{k} $ is the zero extension of $ v_{k} $ to $ L^{\infty} \big(0, T; L^{2}(\mathbb{R}^{n}) \big) $ for $ k \in \mathbb{N} \cup \{ \infty \} $.

Proof. $ L^{\infty} \big(0, T; L^{2}(\Omega^{k}) \big) $ is dual of $ L^{1} \big(0, T; L^{2}(\Omega^{k}) \big) $ for $ k \in \mathbb{N} \cup \{ \infty \} $. We denote $ \bar{v}_{k} $ as the zero extensions of $ v_{k} $ to $ L^{\infty} \big(0, T; L^{2} (\mathbb{R}^{n}) \big) $ for $ k \in \mathbb{N} \cup \{ \infty \} $. Since

by Proposition 2.3 we find that there exists a weakly convergent subsequence, which will be still denoted as $ \{ \bar{v}_{k} \}_{k = 1}^{\infty} $, which converges as

We remark that weak-$ \ast $ convergence was used instead of weak convergence, because $ (L^{\infty})^{\ast} \not = L^{1} $. One can easily check from (1.7) that $ \bar{v}_{\infty} = 0 $ a.e. in $ \mathbb{R}^{n}_{T} \setminus \Omega_{T}^{\infty} $. So the lemma follows by taking $ v_{\infty} = \bar{v}_{\infty} \cdot1_{\Omega_{T}^{\infty} } $.

Now recall the energy estimate (3.2).

Let $ \bar{F}_{k}, \bar{\gamma}_{k}, D\bar{\gamma}_{k} \in L^{p}(\mathbb{R}^{n}_{T}) $ be the zero extension of $ F_{k}, \gamma_{k}, D\gamma_{k} \in L^{p}(\Omega_{T}^{k}) $, respectively. (We remark that $ \bar{\gamma}_{k} $ might not be weakly differentiable in $ \mathbb{R}^{n}_{T} $, but we abuse the notation for the simplicity of the computation.) We apply Lemma 3.1 to (1.9). Then

which implies that

and

Let $ \bar{f}_{k} $, $ \partial_{t} \bar{\gamma}_{k} $, $ \bar{f} $ and $ \partial_{t} \bar{\gamma} $ be the zero extension of $ f_{k}, \partial_{t} \gamma_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) $ and $ f, \partial_{t} \gamma \in L^{p'} \big(0, T; W^{-1, p'}(\Omega) \big) $ in Corollary 3.3 respectively. By Corollary 3.3 and Lemma 3.5, we find from (1.8) that

which implies that

and

So the right-hand side of (3.34) is bounded, and one can apply Aubin-Lions Lemma, Lemma 3.7 and the zero extension to find that there exists a subsequence of $ \{ \bar{u}_{k} \}_{k = 1}^{\infty} $, which will be still denote by $ \{ \bar{u}_{k} \}_{k = 1}^{\infty} $, and $ \bar{u}_{0} \in L^{p} \big(0, T; W^{1, p}_{0} (\mathbb{R}^{n}) \big) \cap L^{\infty}\big(0, T; L^{2}(\mathbb{R}^{n}) \big) $ such that

Here, the compactness method is applied to some ball satisfying $ B \supset \Omega^{k} $ $ (k \in \mathbb{N}) $ and $ B \supset \Omega $ by using the zero extensions.

By (1.10),

for any $ \varphi \in L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) $. Then we see that $ \| \partial_{t} u_{k} \|_{L^{p'} \big(0, T; W^{-1, p'} (\Omega^{k}) \big)} $ is bounded. We denote the zero extension of $ \partial_{t} u_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega^{k}) \big) $ in Corollary 3.3 as $ \partial_{t} \bar{u}_{k} \in L^{p'} \big(0, T; W^{-1, p'}(\mathbb{R}^{n}) \big) $. Then we find from Corollary 3.3 that

So by Lemma 3.6, there exist $ \partial_{t} u_{0} \text{ in } L^{p'} \big(0, T; W^{-1, p'} (\Omega) \big) $ and a subsequence of $ \{ \bar{u}_{k} \}_{k = 1}^{\infty} $, which will be still denoted by $ \{ \bar{u}_{k} \}_{k = 1}^{\infty} $ such that

Here, we denoted the zero extension of $ \partial_{t} u_{0} \in L^{p'} \big(0, T; W^{-1, p'}(\Omega) \big) $ in Corollary 3.3 as $ \partial_{t} \bar{u}_{0} \in L^{p'} \big(0, T; W^{-1, p'}(\mathbb{R}^{n}) \big) $. Define $ u_{0} = \bar{u}_{0} + \gamma $ in $ \Omega_{T} $. Then we have that following lemma. We remark that a different proof is shown in Step 4 in the proof of [30, Lemma 5.1].

Lemma 3.8. For $ u_{0} = \bar{u}_{0} + \gamma $ in $ \Omega_{T} $, we have that

Proof. Let $ \hat{u}_{k} $ be the zero extension of $ \bar{u}_{k} $ from $ \mathbb{R}^{n} \times [0, T] $ to $ \mathbb{R}^{n} \times [-T, T] $, which means that $ \hat{u}_{k} = 0 $ in $ (\mathbb{R}^{n} \times [-T, T]) \setminus (\mathbb{R}^{n} \times [0, T]) $. Also define $ \partial_{t} \hat{u}_{k} $ as

Then we see that $ \partial_{t} \hat{u}_{k} \in L^{p'} \big(-T, T; W^{-1, p'} (\mathbb{R}^{n}) \big) $, because

for any $ \varphi \in C_{c}^{\infty}(\mathbb{R}^{n} \times [-T, T]) $. Here, we used that $ \bar{u}_{k} = 0 $ on $ \mathbb{R}^{n} \times \{ 0 \} $.

By (3.37) and (3.39), there exists a subsequence, which will be still denoted as $ \hat{u}_{k} $ and $ \partial_{t} \hat{u}_{k} $ $ (k \in \mathbb{N}) $, such that

and

for some $ \hat{u}_{0} \in L^{p} \big(-T, T; W^{1, p}_{0} (\mathbb{R}^{n}) \big) \cap L^{\infty}\big(-T, T; L^{2}(\mathbb{R}^{n}) \big) $ and $ \partial_{t} \hat{u}_{0} \in L^{p'} \big(-T, T; W^{-1, p'} (\mathbb{R}^{n}) \big) $. Then by Proposition 2.6, we have that $ \hat{u}_{0} \in C \big([-T, T]; L^{2}(\mathbb{R}^{n}) \big) $, which implies that

where we used that $ \hat{u}_{0} = \bar{u}_{0} $ in $ \mathbb{R}^{n}_{T} $, which holds from (3.37), (3.40) and that $ \hat{u}_{k} $ is the zero extension of $ \bar{u}_{k} $ from $ \mathbb{R}^{n}_{T} $ to $ \mathbb{R}^{n} \times [-T, T] $. Since $ \bar{u}_{0} = u_{0} - \gamma $ in $ \Omega $, we get

Since $ \gamma \in C\big([0, T]; L^{2}(\Omega) \big) $, we find that

and the lemma follows.

Lemma 3.9. For the weak solutions $ u \in \gamma + L^{p} \big(0, T; W^{1, p}_{0}(\Omega) \big) \cap C \big([0, T]; L^{2}(\Omega) \big) $ of (1.6) and $ u_{k} \in \gamma_{k} + L^{p} \big(0, T; W^{1, p}_{0}(\Omega^{k}) \big) \cap C \big([0, T]; L^{2}(\Omega^{k}) \big) $ in (1.10), we have that

and

Moreover, we have that

Proof. Recall from (1.7) that

which implies that

Fix $ \varphi(x) \in C_{c}^{\infty}(\Omega) $ with $ 0 \leq \varphi \leq 1 $, which is independent of $ t $-variable. Choose $ K \in \mathbb{N} $ in (3.43). Test (1.10) by $ \left(\bar{u}_{k} - \bar{u}_{0} \right) \varphi^{p} $ to find that

for any $ k \geq K $. Recall that $ \bar{u}_{k} = u_{k} - \gamma_{k} $, $ \bar{u}_{0} = u_{0} - \gamma $ and $ \varphi \in C_{c}^{\infty}(\Omega) \cap C_{c}^{\infty}(\Omega^{k}) $ for any $ k \geq K $. For $ (\mathop{supp } \varphi)_{T} = \mathop{supp } \varphi \times [0, T] $, we discover that

where

for $ k \geq K $. One can easily check from (3.35) and (3.37) that

By a direct calculation, we have

By (3.35)–(3.37),

We handle $ III_{k} $. By Lemma 2.14,

So by (3.37),

By (3.36) and (3.37),

Since $ \varphi = \varphi(x) $ and $ 0 \leq \varphi \leq 1 $, one can easily show that

because $ \bar{u}_{k} = 0 = \bar{u}_{0} $ on $ \mathbb{R}^{n} \times \{ 0 \} $, which holds from Lemma 3.8. So by (3.44), (3.45), (3.46) and (3.47),

because $ \left \langle a_{k}(Du_{k}, x, t) - a_{k}(Du_{0}, x, t), (Du_{k} - Du_{0})\varphi^{p} \right \rangle \geq 0 $ in $ \mathbb{R}^{n}_{T} \cap (\mathop{supp } \varphi)_{T} $, which implies that

For any $ \kappa \in (0, \kappa_{1}] $, we have from Lemma 2.7 that

So we find that

Since $ \kappa \in (0, \kappa_{1}] $ and $ \varphi \in C_{c}^{\infty}(\Omega) $ were arbitrary chosen, we discover that

So by (3.35),

For any $ U \subset \subset \Omega $, there exists a cut-off function $ \eta \in C_{c}^{\infty} (\Omega) $ such that $ 0 \leq \eta \leq 1 $ in $ \Omega $ and $ \eta = 1 $ on $ U $. Moreover, by (3.42), there exists $ K \in \mathbb{N} $ such that

So by (3.48),

By Corollary 3.3 and (3.39),

for any $ \varphi \in C_{0 }^{\infty} (\Omega_{T}) $.

Now, we show that $ u_{0} $ is the weak solution of (1.6), which implies that $ u = u_{0} $ by the uniqueness. Fix $ \varphi \in C_{0 }^{\infty} (\Omega_{T}) $ and choose $ U \subset \subset \Omega $ with $ \text{supp } \varphi \subset \overline{U_{T}} $. By (3.42), there exists $ K \in \mathbb{N} $ such that $ U \subset \subset \Omega^{k} $ $ (k \geq K) $. We have from (1.10) that

for any $ k \geq K $. So by Lemma 2.10, Lemma 2.14, (3.35), (3.36), (3.50) and (3.51),

We find from Lemma 3.8 that $ u_{0} \in L^{\infty} \big(0, T; L^{2}(\Omega) \big) \cap L^{p} \big(0, T; W^{1, p}_{0} (\Omega) \big) $ is also the weak solution of (1.6). By uniqueness of the weak solution, we find that $ u_{0} = u $, and the lemma follows from (3.37), (3.48) and (3.50).

We next estimate the concentration of $ D\bar{u}_{k} $ near the boundary $ \partial \Omega \times [0, T] $.

Lemma 3.10. For any $ \varphi \in C_{c}^{\infty}(\Omega) $ with $ 0 \leq \varphi \leq 1 $, we have that

Proof. Fix $ \varphi \in C_{c}^{\infty}(\Omega) $ with $ 0 \leq \varphi \leq 1 $. We have from (1.7) that

We next take $ \kappa = \kappa_{1}(n, p, \lambda, \Lambda) $ in Lemma 2.7 to find that

for any $ k \geq K $. In view of (1.2), we discover that

for any $ k \geq K $.

We will estimate the limit superior of the right-hand side of (3.54). We test (1.10) by $ (u_{k}- \gamma_{k}) \left(1-\varphi^{p} \right) $ to find that

where

for any $ k \geq K $.

We estimate the limit of the right-hand side as $ k \to \infty $. Without loss of generality, assume that $ k \geq K $. Then we have from (3.52) that

We first compute the limit of $ I_{k} $. By the triangle inequality,

Since $ \varphi \in C_{c}^{\infty}(\Omega) \cap C_{c}^{\infty}(\Omega^{k}) $, we have from Lemma 2.10, Lemma 2.14 and (3.41) in Lemma 3.9 that

By Lemma 3.9, we have that $ \bar{u}_{k} \to \bar{u} $ in $ L^{p}(\mathbb{R}^{n}_{T}) $. Since $ u_{k} - \gamma_{k} = \bar{u}_{k} $ in $ \Omega_{T}^{k} $ and $ u - \gamma = \bar{u} $ in $ \Omega_{T} $, we find from (3.50) that

Similarly, by the triangle inequality,

So we get from (3.35), Lemma 2.10 and Lemma 2.14 that

and it follows from Lemma 3.9 that

Recall that

Then one can easily check from (3.35), (3.36) and Lemma 3.9 that

Now, we estimate $ IV_{k} $.

Since $ \varphi = \varphi(x) $, $ 0 \leq \varphi \leq 1 $ and $ u_{k} - \gamma_{k} = 0 $ on $ \Omega^{k} \times \{ 0 \} $, we find that

Since $ u_{k} - \gamma_{k} = \bar{u}_{k} $ in $ \Omega_{T}^{k} $ and $ u - \gamma = \bar{u} $ in $ \Omega_{T} $, we find from (3.36) and Lemma 3.9 that

Thus

In view of (3.55), we find from (3.57), (3.58), (3.59) and (3.60) that

By taking $ (u-\gamma) \left(1 - \varphi^{p} \right) $ in (1.6), we get that

Thus

Since $ \bar{u} = u-\gamma $, we find that

Since $ \bar{u}_{k} = u_{k} - \gamma_{k} $, by (3.35), (3.53) and (3.54),

and the lemma follows.

3.1.   Proof of Theorem 1.6

We are ready to prove Theorem 1.6.

Proof of Theorem 1.6. By Lemmas 3.9 and 3.10,

for any $ \varphi \in C_{c}^{\infty}(\Omega) $ with $ 0 \leq \varphi \leq 1 $. Since $ \varphi \in C_{c}^{\infty}(\Omega) $ with $ 0 \leq \varphi \leq 1 $ can be arbitrary chosen in the above estimates, one can choose a sequence of monotone increasing functions in $ C_{c}^{\infty}(\Omega) $ which converges to $ 1 $ a.e. in $ \Omega $. Then by Lebesgue's dominated convergence theorem, we get

This contradicts (3.1). So we find that (1.11) holds.

Acknowledgments

Y. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. NRF-2020R1C1C1A01013363). S. Ryu was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01014310). P. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2020R1I1A1A01066850). The authors would like to thank the referee for the careful reading of this manuscript and for offering valuable comments.

Conflict of interest

The authors declare no conflict of interest.