Optimal $C^\infty$-approximation of functions with exponentially or sub-exponentially integrable derivative

Abstract

We discuss Meyers-Serrin’s type results for smooth approximations of functions $b=b(t,x): \mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {R}^m$, with convergence of an energy of the form

$$\begin{aligned} \int _{\mathbb {R}}\int _{\mathbb {R}^n} w(t,x) \varphi \left( |Db(t,x)|\right) \textrm{d} x \textrm{d} t\,, \end{aligned}$$

where $w>0$ is a suitable weight function, and $\varphi :[0,\infty )\rightarrow [0, \infty )$ is a convex function with $\varphi (0)=0$ having exponential or subexponential growth.

1 Introduction and results

In this note we deal with the approximation of functions $b=b(t,x):I\times \Omega \rightarrow \mathbb {R}^m$ by smooth ones, where $I\subset \mathbb {R}$ and $\Omega \subset \mathbb {R}^n$ are an open interval and an open set, respectively. Our main motivation comes from [4], where $m=n$ and b is a possibly nonautonomous vector field. In that paper we are dealing with a priori upper bounds for Sobolev norms of the flow of b by means of quantities $N_{\varphi ,w}(|Db|)$ that depend on the spatial derivative Db of b. The quantities $N_{\varphi ,w}(|Db|)$ are energies of the form

$$\begin{aligned} N_{\varphi ,w}(|Db|):=\, \int _I\int _\Omega w(t,x) \varphi \left( |Db(t,x)|\right) \,\textrm{d}x \,\textrm{d}t, \end{aligned}$$

where $w>0$ is a suitable weight function (related to $\textrm{dist}(x,\partial \Omega )$, or to the length of the maximal interval of the ODE associated to b) and $\varphi :[0,\infty )\rightarrow [0,\infty )$ is a convex function with $\varphi (0)=0$ having exponential or sub-exponential growth, the model case being $\exp _*(t):=\exp (t)-1$. When one tries to extend the a priori estimates from the case of smooth vector fields b to those having only a Sobolev spatial regularity, one faces the difficulty of passing the quantity $N_{\varphi ,w}$ to the limit.

In this context, if $\varphi$ had polynomial growth, a weighted and t-dependent version of the celebrated Meyers-Serrin Theorem [9] would be applicable, providing even a smooth approximation $(b_h)_h$ with $N_{\varphi ,w}(|D(b_h-b)|)\rightarrow 0$. In general, as the discussion below shows, this kind of approximation fails when $\varphi$ does not satisfy a doubling condition. However, we realized that the exponential (or subexponential) case is a borderline one. Indeed, thanks to the weak subadditivity condition

$$\begin{aligned} \varphi (a+b)\le \,k\,[(1+\varphi (b))\,\varphi (a)+\varphi (b)] \end{aligned}$$

we are able to prove convergence of the energy $N_{\varphi ,w}$ and density of smooth functions with respect to modular convergence when $\varphi$ is strictly convex. This kind of convergence in energy, though weaker than convergence with respect to Luxenburg norm (or modular convergence, when $\varphi$ is not strictly convex), should be compared with the theory of BV functions, where smooth functions are not dense in BV norm, but dense in energy. Moreover, this convergence will be sufficient to pass our a priori estimates to the limit in the paper [4].

In the classical setting of Orlicz spaces (see Sect. 2) the weighted $\varphi$-energy $N_{\varphi ,w}$ is called a modular and we put some of our results in this context. The main approximation result of the note reads as follows.

Theorem 1

Let $I\subset \mathbb {R}$ be an open interval and $\Omega \subset \mathbb {R}^n$ an open set. Let $w:\,I\times \Omega \rightarrow (0,\infty )$ be a Borel function uniformly bounded from above and from below on compact subsets of $I\times \Omega$. Let $\varphi :\,[0,\infty )\rightarrow [0,\infty )$ be a convex function satisfying $\varphi (0)=\,0$ and and for which there exists a positive constant $k_\varphi$ such that

$$\begin{aligned} \varphi (a+b)\le \,k_\varphi [\varphi (a)\,\varphi (b)+\,\varphi (a)+\varphi (b)]\qquad \text {for all} a,\,b\in [0,\infty ). \end{aligned}$$

(1)

Let $b\in L^1_\textrm{loc}(I;W^{1,1}_\textrm{loc}(\Omega ;\mathbb {R}^m))\cap C^0(I\times \Omega ;\mathbb {R}^m)$ satisfy

$$\begin{aligned} N_{\varphi ,w}(|Db|)<\infty . \end{aligned}$$

(2)

Then there exist $b_h\in C^\infty (I\times \Omega ;\mathbb {R}^m)$ satisfying

$$\begin{aligned} b_h\rightarrow b \text { in }L^1_\textrm{loc}(I\times \Omega ;\mathbb {R}^m), \qquad Db_h\rightarrow Db\text { in }L^1_\textrm{loc}(I\times \Omega ;\mathbb {R}^{nm}), \end{aligned}$$

(3)

and

$$\begin{aligned} w\varphi \left( |Db_h|\right) \rightarrow w\varphi \left( |Db|\right) \text { in }L^1(I\times \Omega ). \end{aligned}$$

(4)

In particular,

$$\begin{aligned} \lim _{h\rightarrow \infty } N_{\varphi ,w}(|Db_h|) = N_{\varphi ,w}(|Db|) . \end{aligned}$$

(5)

Besides the model case of $\varphi =\exp _*$, we are able to consider the functions

$$\begin{aligned} \exp _{\gamma ,\tau }^*(t):=\,\exp _{\gamma ,\tau }(t)-1, \end{aligned}$$

(6)

where $0\le \,\gamma \le \,1$, $\tau >0$ and

$$\begin{aligned} \exp _{\gamma ,\tau }(t):=\exp \left( \displaystyle \frac{t}{(\log (t+\tau ))^\gamma }\right) . \end{aligned}$$

(7)

It is easy to see that a convex function $\varphi$ with polynomial growth satisfies (1), see Remark 6. We will show in Lemma 21 that $\exp _{\gamma ,\tau }^*$ satisfies the conditions in Theorem 1 for $\varphi$ with $k_\varphi =1$, if $\tau$ is sufficiently large. The functions $\exp _{\gamma ,\tau }$, though convex, do not have null derivative at 0, and therefore do not fit exactly in the theory of N-functions. Therefore, in order to provide a bridge with the theory of N-functions of Orlicz spaces, we will also consider the modified functions

$$\begin{aligned} \widetilde{\exp }_{\gamma ,\tau }(t):=\,\exp _{\gamma ,\tau }(t)-1-\frac{t}{(\log \tau )^\gamma }= \exp _{\gamma ,\tau }^*(t)-\frac{t}{(\log \tau )^\gamma }\,, \end{aligned}$$

(8)

which are indeed N-functions and can be treated it by comparison with $\exp _{\gamma ,\tau }^*$.

Corollary 2

Let $w:\,I\times \Omega \rightarrow (0,\infty )$ be a Borel function uniformly bounded from above and from below on compact subsets of $I\times \Omega$. Let $b\in L^1_\textrm{loc}(I;W^{1,1}_\textrm{loc}(\Omega ;\mathbb {R}^m))\cap C^0(I\times \Omega ;\mathbb {R}^m)$ satisfy (2) with

$$\begin{aligned} \varphi =\,\widetilde{\exp }_{\gamma ,\tau }\, \text {with } \tau \text { sufficiently large} \end{aligned}$$

(9)

and

$$\begin{aligned} \text {either }w|Db|\in L^1(I\times \Omega )\text {, or }w\in L^1(I\times \Omega ). \end{aligned}$$

(10)

Then there exist $b_h\in C^\infty (I\times \Omega ;\mathbb {R}^m)$ satisfying (3), (4) and (5).

Notice that, when the functions $\varphi$ we are dealing with have a more than polynomial growth at infinity and the weight function w is uniformly bounded from 0 on compact subsets, Corollary 2, the Sobolev Embedding Theorem grants continuity of b with respect to the spatial variable. But, in the proof of Theorem 1, it seems that the continuity of b with respect to t is also needed (cf. the estimate of the term $z_\delta$). However, if we assume the weight w to be time-independent, we can adapt the proof of Theorem 1 to drop the continuity assumption on b, and we obtain the following extension of Corollary 2.

Theorem 3

Let $w:\,\Omega \rightarrow (0,\infty )$ be a Borel function uniformly bounded from above and from below on compact subsets of $\Omega$. Assume that either $\varphi =\exp ^*_{\gamma ,\tau }$ or $\varphi =\widetilde{\exp _{\gamma ,\tau }}$ and (10) holds, with $\tau$ sufficiently large and let $b\in L^1_\textrm{loc}(I;W^{1,1}_\textrm{loc}(\Omega ;\mathbb {R}^m))$ satisfy (2). Then there exist $b_h\in C^\infty (I\times \Omega ;\mathbb {R}^m)$ satisfying (3), (4) and (5).

When the function $\varphi$ is strictly convex, from the $\varphi$-energy convergence of the Jacobian matrices of autonomous vector fields, one can obtain the modular convergence (see Definition 10).

Theorem 4

Let $\varphi :[0,\infty )\rightarrow [0,\infty )$ be a strictly convex function, and assume that $b_h\rightarrow b$ in $L^1_\textrm{loc}(\Omega ;\mathbb {R}^m)$ with

$$\begin{aligned} \int _\Omega \varphi (|Db_h|)\,\textrm{d}x\rightarrow \int _\Omega \varphi (|Db|)\,\textrm{d}x<\infty . \end{aligned}$$

(11)

Then

$$\begin{aligned} \int _\Omega \varphi \left( \frac{|Db_h-D b|}{2}\right) \,\textrm{d}x\rightarrow 0. \end{aligned}$$

(12)

Finally, if $\varphi =\,\widetilde{\exp }_{\gamma ,\tau }$, being $\widetilde{\exp }_{\gamma ,\tau }$ a N-function (see Lemma 21.(ii)), we can set our result in the classical setting of Orlicz-Sobolev spaces (see Sect. 2). Therefore, an immediate consequence of Theorem 3 is the following approximation result in the Orlicz-Sobolev class $W^1 K_{\widetilde{\exp }_{\gamma ,\tau }}(\Omega )$.

Theorem 5

Let $\varphi =\,\widetilde{\exp }_{\gamma ,\tau }$ be the N-function in (8) with $\tau$ given by Lemma 21 and $u\in W^1 K_{\widetilde{\exp }_{\gamma ,\tau }}(\Omega )$. Suppose that

$$\begin{aligned} \text {either }|Du|\in L^1(\Omega )\text {, or }\Omega \text { has finite measure. } \end{aligned}$$

(13)

Then there exists $(u_h)_h\subset C^\infty (\Omega )\cap W^1 K_{\widetilde{\exp }_{\gamma ,\tau }}(\Omega )$ such that $(u_h)_h$ is mean convergent to u (with respect to the modular $N_{\widetilde{\exp }_{\gamma ,\tau }}$) and $(|Du_h|)_h$ is ${\widetilde{\exp }_{\gamma ,\tau }}$-energy convergent to |Du|, that is,

$$\begin{aligned} \lim _{h\rightarrow \infty }N_{\widetilde{\exp }_{\gamma ,\tau }}\left( u_h-u\right) =\,0\text { and }\lim _{h\rightarrow \infty }N_{\widetilde{\exp }_{\gamma ,\tau }}\left( |Du_h|\right) = N_{\widetilde{\exp }_{\gamma ,\tau }}\left( |Du|\right) . \end{aligned}$$

(14)

To our knowledge, the previous result does not seem be a consequence of the well-known results about approximation by smooth functions in Orlicz-Sobolev spaces (see Sect. 2.6), even in the classical case with $\gamma =0$.

2 Recalls of some density results of smooth functions in Orlicz and Orlicz-Sobolev spaces

We will quickly recall here the notions of Orlicz and Orlicz-Sobolev spaces and some their main properties. In particular, we will focus on the main density results of smooth functions in Orlicz and Orlicz-Sobolev spaces. We will mainly use the notation from [1, Ch. VIII].

2.1 N-functions

A function $\varphi :[0,\infty )\rightarrow [0,\infty )$ is called a N-function, if

$$\begin{aligned} \varphi (t):=\,\int _0^t a(s)\,\,\textrm{d}s\text { if }t\ge \,0, \end{aligned}$$

with $a:\,[0,\infty )\rightarrow [0,\infty )$ satisfying:

• $a(0)=\,0$, $a(t)>\,0$ if $t>\,0$, and $\lim _{t\rightarrow \infty }a(t)=\,\infty$;

• a is nondecreasing, that is, if $t\ge \,s\ge \,0$, then $a(t)\ge \,a(s)$;

• a is right continuous, that is, if $t\ge \,0$, then $\lim _{s\rightarrow t^+}a(s)=\,a(t)$.

Given a N-function $\varphi$ and $\lambda >\,0$, we denote by $\varphi _\lambda :[0,\infty )\rightarrow [0,\infty )$ the function

$$\begin{aligned} \varphi _\lambda (t):=\,\varphi \left( \frac{t}{\lambda }\right) \text { if }t\ge \,0, \end{aligned}$$

which is still a N-function.

A function $\varphi$ is said to satisfy a global $\Delta _2$-condition if there exists $k>\,0$ such that

$$\begin{aligned} \varphi (2t)\le \,k\,\varphi (t)\text { for each }t\ge \,0. \end{aligned}$$

A function $\varphi$ is said to satisfy a $\Delta _2$-condition near infinity if there exist $k,t_0>\,0$ such that

$$\begin{aligned} \varphi (2t)\le \,k\,\varphi (t)\text { for each }t\ge \,t_0. \end{aligned}$$

Remark 6

Observe that a convex function $\varphi$ satisfying a global $\Delta _2$-condition trivially fulfills condition (1). Indeed, by the convexity and $\Delta _2$-condition, we can get the following estimate

$$\begin{aligned} \varphi (a+b)\le \,\frac{1}{2}\varphi (2a)+ \frac{1}{2}\varphi (2b)\le \,\frac{k}{2}\left( \varphi (a)+\varphi (b)\right) \text { for each }a,\,b\in \mathbb {R}. \end{aligned}$$

Given $\Omega \subset \mathbb {R}^n$ and a N-function $\varphi$, a pair $(\varphi ,\Omega )$ is said to be $\Delta$ -regular if

• $\varphi$ satisfies a global $\Delta _2$-condition, or

• $\varphi$ satisfies a $\Delta _2$-condition near infinity and $\Omega$ has finite measure.

2.2 The Orlicz class $K_\varphi (\Omega )$

Let $\Omega \subset \mathbb {R}^n$ be an open set and let $\varphi$ be a N-function. The Orlicz class $K_\varphi (\Omega )$ is the set of all (equivalence classes modulo equality a.e. on $\Omega$ of) measurable functions $u:\Omega \rightarrow \mathbb {R}$ such that

$$\begin{aligned} N_\varphi (u):=\,\int _\Omega \varphi (|u(x)|)\,\,\textrm{d}x<\,\infty . \end{aligned}$$

In the theory of modular spaces, the map $u\mapsto N_\varphi (u)$ is called a modular ( [10, pg. 82]). A comprehensive account of modular function spaces can be found in [7]. We treat the case of real-valued functions for simplicity, but all results have an obvious extension to the case of $\mathbb {R}^m$-valued maps.

Let us recall some properties of the Orlicz class $K_\varphi (\Omega )$.

Proposition 7

Given an open set $\Omega \subset \mathbb {R}^n$ and a N-function $\varphi$, the following statements hold:

• (i) $K_\varphi (\Omega )$ is a convex set of measurable functions.

• (ii) $K_{\varphi _\lambda }(\Omega )\supseteq K_\varphi (\Omega )$ if $\lambda \ge \,1$ and $K_{\varphi _\lambda }(\Omega )\subseteq K_\varphi (\Omega )$ if $\lambda \le \,1$, where $\varphi _\lambda (t):=\varphi (t/\lambda )$ is a N-function for all $\lambda >0$.

• (iii) If $f,\,g\in K_{\varphi }(\Omega )$, then $f+g\in K_{\varphi _2}(\Omega )$ and

  $$\begin{aligned} N_{\varphi _2}(f+g)\le \, \frac{1}{2} N_\varphi (f)+\frac{1}{2} N_\varphi (g). \end{aligned}$$

• (iv) If $f\in K_{\varphi }(\Omega )$ and $\lambda >\,0$, then $\lambda f\in K_{\varphi _\lambda }(\Omega )$.

• (v) If $\Omega$ has finite measure, then

  $$\begin{aligned} L^\infty (\Omega )\subset K_\varphi (\Omega ) \subsetneq L^1(\Omega ). \end{aligned}$$

• (vi) If $\Omega$ has finite measure, then for every $u\in L^1(\Omega )$ there is a N-function $\varphi$ such that $u\in K_\varphi (\Omega )$.

Proof

Properties (i), (ii), (iii) and (iv) are immediate consequences of the definition of $K_\varphi (\Omega )$ and the convexity of $\varphi$. For the proof of properties (v) and (vi) see, for instance, [8]. $\square$

Lemma 8

( [1, Lem. 8.8] or [8, Ch. III, Th. 8.2]) $K_\varphi (\Omega )$ is a vector space if and only if $(\varphi ,\Omega )$ is $\Delta$ -regular.

2.3 The Orlicz space $L_\varphi (\Omega )$

The Orlicz space $L_\varphi (\Omega )$ is defined to be the linear hull of the Orlicz class $K_\varphi (\Omega )$, that is the smallest vector subspace of $L^1_\textrm{loc}(\Omega )$ containing $K_\varphi (\Omega )$. It is easy to see that, since $K_\varphi (\Omega )$ is convex, one has

$$\begin{aligned} L_\varphi (\Omega ):=\,\left\{ \lambda \,u:\,\lambda \in \mathbb {R},\,u\in K_\varphi (\Omega )\right\} . \end{aligned}$$

Moreover, from Lemma 8, $K_\varphi (\Omega )=L_\varphi (\Omega )$ if and only if $(\varphi ,\Omega )$ is $\Delta$ -regular.

We can endow $L_\varphi (\Omega )$ with the following norm, called Luxemburg norm,

$$\begin{aligned} \Vert u\Vert _\varphi =\, \Vert u\Vert _{\varphi ,\Omega }:=\,\inf \left\{ \lambda >\,0:\,\int _\Omega \varphi \left( \frac{|u(x)|}{\lambda }\right) \,\,\textrm{d}x\le \,1\right\} . \end{aligned}$$

Theorem 9

( [1, Thm. 8.10]) $(L_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )$ is a Banach space.

2.4 Convergences in $L_\varphi (\Omega )$

The typical convergences that apply in Orlicz spaces are the following.

Definition 10

A sequence of functions $(u_h)_h\subset L_\varphi (\Omega )$ is said to be norm convergent to $u\in L_\varphi (\Omega )$ if

$$\begin{aligned} \Vert u_h-u\Vert _\varphi \rightarrow 0\text { as }h\rightarrow \infty . \end{aligned}$$

A sequence of functions $(u_h)_h\subset L_\varphi (\Omega )$ is said to be modular convergent to $u\in L_\varphi (\Omega )$ if there exists $\lambda >\,0$ such that

$$\begin{aligned} N_\varphi \left( \frac{u_h-u}{\lambda }\right) \rightarrow 0\text { as }h\rightarrow \infty . \end{aligned}$$

(15)

If $\lambda =1$ in (15), $(u_h)_h$ is said to be mean convergent to $u\in L_\varphi (\Omega )$. A sequence of functions $(u_h)_h\subset K_\varphi (\Omega )$ is said to be $\varphi$ -energy convergent to $u\in K_\varphi (\Omega )$ if

$$\begin{aligned} N_\varphi (u_h)\rightarrow N_\varphi (u)\text { as }h\rightarrow \infty . \end{aligned}$$

(16)

Norm and modular convergences are classical in the theory of Orlicz spaces (see, for instance, [1, 8]). We do not know whether the $\varphi$-energy convergence has been already named in the literature.

The following implications between norm, mean, modular and $\varphi$-energy convergence hold.

Proposition 11

Let $(u_h)_h$ and u be in $L_\varphi (\Omega )$.

• (i) Suppose that $(u_h)_h$ is norm convergent to u. Then it is also mean convergent. The converse implication in general does not hold. It holds if $(\varphi ,\Omega )$ is $\Delta$ -regular.

• (ii) Suppose that $(\varphi ,\Omega )$ is $\Delta$-regular, $\varphi$ is strictly convex, $u_h\rightarrow u$ a.e. in $\Omega$ and $(u_h)_h$ is $\varphi$-energy convergent to u. Then $(u_h)_h$ is norm convergent to u.

• (iii) $(u_h)_h$ is norm convergent to u if and only if, for each $\lambda >\,0$,

  $$\begin{aligned} N_\varphi \left( \frac{u_h-u}{\lambda }\right) \rightarrow 0\text { as }h\rightarrow \infty . \end{aligned}$$

• (iv) Suppose that $(\varphi ,\Omega )$ is $\Delta$ -regular and $(u_h)_h\subset K_\varphi (\Omega )$ is mean convergent to $u\in K_\varphi (\Omega )$. Then $(u_h)_h$ is $\varphi$-energy convergent to u.

• (v) Suppose that $(2u_h)_h\subset K_{\varphi }(\Omega )$ is mean convergent to $2u\in K_{\varphi }(\Omega )$ (with respect to the modular $N_{\varphi }$). Then $(u_h)_h\subset K_{\varphi }(\Omega )$, $u\in K_{\varphi }(\Omega )$ and $(u_h)_h$ is $\varphi$-energy convergent to u.

• (vi) Suppose that $2u\in K_\varphi (\Omega )$ and $(u_h)_h$ is norm convergent to u. Then $u_h\in K_\varphi (\Omega )$ for h large and $(u_h)$ is also $\varphi$-energy convergent to u.

Proof

(i) and (ii) are proven in [10, Chap. III, Sect. 3.4, Thm. 12]. The proof of (iii) is somehow elementary, see for instance [2, Lem. 2.7] and [7, pg. 4].

We prove (iv) when $\varphi$ satisfies a global $\Delta _2$-condition: in the other case, when $\Omega$ has finite measure and the $\varphi$ satisfies a $\Delta _2$-condition near infinity, has a similar proof. From the assumptions, the sequence $(\varphi (|u_h-u|))_h\subset L^1(\Omega )$ converges in $L^1(\Omega )$ to 0. Thus, up to a subsequence, we can assume that

$$\begin{aligned} \varphi (|u_h-u|)\rightarrow 0\text { a.e. in } \Omega ,\text { as }h\rightarrow \infty . \end{aligned}$$

Since $\varphi$ is a N-function, then $\varphi :\,[0,\infty )\rightarrow [0,\infty )$ is bijective and $\varphi ^{-1}:\,[0,\infty )\rightarrow [0,\infty )$ is still continuous. Thus, we also get that

$$\begin{aligned} |u_h-u|=\,\varphi ^{-1}(\varphi (|u_h-u|))\rightarrow \varphi ^{-1}(\varphi (0))=\,0\text { a.e.~in } \Omega ,\text { as }h\rightarrow \infty . \end{aligned}$$

(17)

From the convexity of $\varphi$ and the global $\Delta _2$-condition of $\varphi$, it follows that

$$\begin{aligned} \begin{aligned} \varphi (|u_h|)&\le \,\frac{1}{2}\varphi (2|u_h-u|)+ \frac{1}{2}\varphi (2|u|)\,\\&\le \,\frac{k}{2}\left( \varphi (|u_h-u|)+\varphi (|u|)\right) . \end{aligned} \end{aligned}$$

(18)

By (17) and (18), we can apply Vitali’s convergence theorem and then

$$\begin{aligned} \varphi (|u_h|)\rightarrow \varphi (|u|)\text { in }L^1(\Omega )\text {, as }h\rightarrow \infty . \end{aligned}$$

Thus (16) follows.

For (v), we get at once that $(u_h)_h\subset K_{\varphi }(\Omega )$ and $u\in K_{\varphi }(\Omega )$, because $\varphi$ is increasing. We can show (17) as in the proof of claim (iv), and the convexity of $\varphi$ implies

$$\begin{aligned} \varphi (|u_h|) \le \,\frac{1}{2}\varphi (2|u_h-u|)+ \frac{1}{2}\varphi (2|u|) \end{aligned}$$

Thus, applying Vitali’s convergence theorem, we still get (16).

Finally, we prove (vi). From the norm convergence and (iii), we can infer that, up to a subsequence,

$$\begin{aligned} \varphi (|u_h-u|)\rightarrow 0\text { a.e. in }\Omega \text {, as }h\rightarrow \infty , \end{aligned}$$

and

$$\begin{aligned} \varphi (2|u_h-u|)\rightarrow 0\text { in } L^1(\Omega )\text {, as }h\rightarrow \infty . \end{aligned}$$

We can show again (17) as in claim (iv) and $\varphi (|u_h|)\le \frac{1}{2}\varphi (2|u_h-u|)+ \frac{1}{2}\varphi (2|u|)$ from the convexity of $\varphi$. Thus, applying Vitali’s convergence theorem, we get (16). $\square$

Example 12

From items (iv) and (v) of Proposition 11, one could get the wrong impression that mean convergence implies $\varphi$-energy convergence. We show that this is not the case if $(\varphi ,\Omega )$ is not $\Delta$-regular: for $\Omega =\,(0,1)$ and $\varphi =\,\widetilde{\exp }_0$ (cf. (8)), we give a sequence of functions $u_h\in K_\varphi ((0,1))$ that is mean convergent to $u\in K_\varphi ((0,1))$, but that is not $\varphi$-energy convergent.

Let $f_h,f:(0,1)\rightarrow \mathbb {R}$ be the functions

$$\begin{aligned} \displaystyle { f_h(x):= {\left\{ \begin{array}{ll} \frac{2\sqrt{h}}{\log h}&{}\text { if }0<\,x<\,\frac{1}{h}\\ \frac{1}{\log h}\frac{1}{\sqrt{x}}&{}\text { if }\frac{1}{h}\le \,x<\,1 \end{array}\right. } },\quad f(x):=\frac{1}{\sqrt{x}}. \end{aligned}$$

Direct computations show that

$$\begin{aligned} \int _0^1 f(x) \,\textrm{d}x&= 2 ,&\int _0^1 \log (f(x)) \,\textrm{d}x&= \frac{1}{2} , \\ \int _0^1 f_h(x) \,\textrm{d}x&\overset{h\rightarrow \infty }{\longrightarrow }0 ,&\int _0^1 \log (1+f_h) \,\textrm{d}x&\overset{h\rightarrow \infty }{\longrightarrow }0 , \\ \int _0^1 f(x)\,f_h(x) \,\textrm{d}x&= \frac{4}{\log (h)} + 1 \overset{h\rightarrow \infty }{\longrightarrow }1 .\\ \end{aligned}$$

Define

$$\begin{aligned} u:= \log (f) \quad \text { and }\quad u_h:= \log (f) + \log (1+f_h). \end{aligned}$$

Then, for $\varphi (s) = \exp (s)-1-s$, we have

$$\begin{aligned} N_\varphi (u)&= \int _0^1 f(x) \,\textrm{d}x - 1 - \int _0^1 \log (f(x)) \,\textrm{d}x , \\ N_\varphi (u_h)&= \int _0^1 f(x) \,\textrm{d}x - 1 - \int _0^1 \log (f(x)) \,\textrm{d}x \\&\qquad + \int _0^1 f(x)\,f_h(x) \,\textrm{d}x - \int _0^1 \log (1+f_h) \,\textrm{d}x , \\ N_\varphi (u_h-u)&= \int _0^1 f_h(x) \,\textrm{d}x - \int _0^1 \log (1+f_h(x)) \,\textrm{d}x . \end{aligned}$$

We conclude that $u,u_h \in K_{\varphi }((0,1))$, $N_\varphi (|u-u_h|)\rightarrow 0$ but

$$\begin{aligned} N_\varphi (|u_h|) - N_\varphi (|u|) = \int _0^1 f(x)\,f_h(x) \,\textrm{d}x - \int _0^1 \log (1+f_h) \,\textrm{d}x \overset{h\rightarrow \infty }{\longrightarrow }1, \end{aligned}$$

that is, $u_h$ is not $\varphi$-energy convergent to u. Notice that the key fact is that $f_h\rightarrow 0$ in $L^1((0,1))$ but $f\cdot f_h\not \rightarrow 0$. Let us also observe that $(2u_h)_h\subset K_{\varphi }(\Omega )$, but neither $2u\in K_{\varphi }(\Omega )$ nor $(2u_h)_h$ is mean convergent to 2u with respect to $N_{\varphi }$.

2.5 The vector space $E_\varphi (\Omega )$

Let $E_\varphi (\Omega )$ denote the closure in $(L_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )$ of the space of functions u which are bounded in $\Omega$ with bounded support in $\Omega$.

One can see ( [1, Sect. 8.14]) that $E_\varphi (\Omega )\subset \,K_\varphi (\Omega )$ and that, if $(\varphi ,\Omega )$ is $\Delta$ -regular, then

$$\begin{aligned} E_\varphi (\Omega )=\,K_\varphi (\Omega )=\,L_\varphi (\Omega ). \end{aligned}$$

Moreover the following characterization of $E_\varphi (\Omega )$ holds ([A, Lemma 8.15]).

Lemma 13

$E_\varphi (\Omega )$ is the maximal linear subspace of $K_\varphi (\Omega )$.

Corollary 14

If $(\varphi ,\Omega )$ is not $\Delta$ -regular, it holds that

$$\begin{aligned} E_\varphi (\Omega )\subsetneq \,K_\varphi (\Omega ) \subsetneq L_\varphi (\Omega ). \end{aligned}$$

Proof

By Lemma 8, $K_\varphi (\Omega )$ cannot be a vector space. Thus, by Lemma 13, we get the desired conclusions. $\square$

Let us now recall some density results in $(E_\varphi (\Omega ),\Vert \cdot \Vert _A)$.

Theorem 15

( [1, Thm. 8.20]) Let $\Omega \subset \mathbb {R}^n$ be an open set and let $\varphi$ be a N-function.

• (i) $C^\infty _c(\Omega )$ are dense in $(E_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )$.

• (ii) $(E_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )$ is separable.

• (iii) Let us extend $u\in E_\varphi (\Omega )$ to the whole $\mathbb {R}^n$ so as to vanish outside $\Omega$ and let $(\rho _\varepsilon )_\varepsilon$ be a family of mollifiers on $\mathbb {R}^n$. Then

  $$\begin{aligned} \rho _\varepsilon *u\rightarrow u \text { in }(E_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )\text {, as } \varepsilon \rightarrow 0. \end{aligned}$$

An immediate consequence of Theorem 15 is that, if $(\varphi ,\Omega )$ is not $\Delta$-regular, then $C^0_c(\Omega )$ is not dense in $(L_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )$. In fact, one can prove the following stronger result:

Theorem 16

( [8, Chap. II, Thm. 10.2]) If the pair $(\varphi ,\Omega )$ is not $\Delta$-regular, then $(L_\varphi (\Omega ),\Vert \cdot \Vert _\varphi )$ is not separable.

Let us also point out some density results in $K_\varphi (\Omega )$ with respect to the modular convergence.

Theorem 17

Let $\Omega \subset \mathbb {R}^n$ be an open set and let $\varphi$ be a N-function.

• (i) The set of bounded functions on $\Omega$ contained in $K_\varphi (\Omega )$ with bounded support is dense in $K_\varphi (\Omega )$ with respect to the mean convergence, that is, for each $u\in K_\varphi (\Omega )$ there exists a sequence of bounded functions $(u_h)_h\subset K_\varphi (\Omega )$ such that

  $$\begin{aligned} N_\varphi (u_h-u)\rightarrow 0\text {, as }h\rightarrow \infty . \end{aligned}$$

• (ii) $C^0_c(\Omega )$ is dense in $K_\varphi (\Omega )$ with respect to the modular convergence with $\lambda =4$. More precisely, for each $u\in K_\varphi (\Omega )$, there is a sequence $(u_h)_h\subset C^0_c(\Omega )$ such that

  $$\begin{aligned} N_\varphi \left( \frac{u_h-u}{4}\right) \rightarrow 0\text {, as }h\rightarrow \infty . \end{aligned}$$

Proof

The proof of part (i) can be found in [8, Chap. II, pg. 77] or [1, Sect. 8.14].

To prove part (ii), given $u\in K_\varphi (\Omega )$ and $\epsilon >0$, we first notice that, by a standard truncation argument in $\Omega$, there is a function ${\tilde{u}}\in K_\varphi (\Omega )$ with support compactly contained in $\Omega$ and with $N_{\varphi }(u-{\tilde{u}})<\epsilon$.

Next, let ${\tilde{u}}_k:=\max \{-k,\min \{{\tilde{u}},k\}\}$ be the standard truncation of u, $F_k:=\{|{\tilde{u}}|>k\}$ and $f\in C^0_c(\Omega )$ with $\sup |f|\le k$. We estimate

$$\begin{aligned} N_{\varphi _2}({\tilde{u}}-f)&= \int _{\Omega \setminus F_k}\varphi \left( \frac{|{\tilde{u}}_k-f|}{2}\right) \,\textrm{d}x+\int _{F_k} \varphi \left( \frac{|{\tilde{u}}-f|}{2}\right) \,\textrm{d}x \\&\le \int _{\Omega \setminus F_k}\varphi \left( \frac{|{\tilde{u}}_k-f|}{2}\right) \,\textrm{d}x + \frac{1}{2}\int _{F_k} \varphi (|{\tilde{u}}|)\,\textrm{d}x + \frac{1}{2}\int _{F_k} \varphi (|f|)\,\textrm{d}x \\&\le \int _{\Omega \setminus F_k}\varphi \left( \frac{|{\tilde{u}}_k-f|}{2}\right) \,\textrm{d}x + \int _{F_k} \varphi (|{\tilde{u}}|)\,\textrm{d}x . \end{aligned}$$

Now, since $\int _\Omega \varphi (|{\tilde{u}}|)\,\textrm{d}x<\infty$, we can choose k so large that the second integral is smaller than $\epsilon /2$. Since ${\tilde{u}}_k$ has compact support and thanks to Lusin’s theorem, we can find $f\in C^0_c(\Omega )$ with $|f|\le k$ and the Lebesgue measure of $\{x\in \Omega :\ {\tilde{u}}_k(x)\ne f(x)\}$ sufficiently small, in such a way that also the first integral gives a contribution smaller than $\epsilon /2$.

In conclusion, for every $\epsilon >0$ we have $f\in C^0_c(\Omega )$ such that

$$\begin{aligned} N_{\varphi _4}(u-f) \le \frac{1}{2} (N_{\varphi _2}(u-{\tilde{u}}) + N_{\varphi _2}({\tilde{u}} - f) ) \le \frac{1}{2} (N_{\varphi }(u-{\tilde{u}})/2 + \epsilon ) \le \epsilon . \end{aligned}$$

$\square$

2.6 Orlicz-Sobolev spaces and density results of smooth functions.

Given a N-function $\varphi$, the Orlicz-Sobolev vector space $W^1L_\varphi (\Omega )$ consists of those (equivalence classes of) functions $u\in L_\varphi (\Omega )\cap W^{1,1}_{loc}(\Omega )$ whose weak derivatives $D_iu\in L_\varphi (\Omega )$ for each $i=1,\ldots ,n$. The vector space $W^1E_\varphi (\Omega )$ and the convex set $W^1K_\varphi (\Omega )$ are defined in analogous fashion. Obviously

$$\begin{aligned} W^1E_\varphi (\Omega )\subset W^1K_\varphi (\Omega )\subset W^1L_\varphi (\Omega ). \end{aligned}$$

It is easy to see (see, for instance, [1, §8.27]) that $W^1L_\varphi (\Omega )$ is a Banach space with respect to the norm

$$\begin{aligned} \Vert u\Vert _{1,\varphi }:=\,\max \{\Vert u\Vert _{\varphi },\Vert D_1u\Vert _{\varphi },\ldots , \Vert D_nu\Vert _{\varphi }\} . \end{aligned}$$

Notice also that, since

$$\begin{aligned} \max _i|D_iu| \le |Du| \le \,\sum _{i=1}^n|D_iu| \text { a.e. in }\Omega , \end{aligned}$$

and $L_\varphi (\Omega )$ is a linear space, an equivalent norm on $W^1L_\varphi (\Omega )$ is given by.

$$\begin{aligned} \Vert u\Vert _{\varphi }+\Vert |Du|\Vert _{\varphi }. \end{aligned}$$

Observe that $W^1E_\varphi (\Omega )$ turns out to be a closed subspace of $W^1L_\varphi (\Omega )$. Moreover $W^1E_\varphi (\Omega )$ coincides with $W^1L_\varphi (\Omega )$ if and only if $(\varphi ,\Omega )$ is $\Delta$ -regular. Notice also that, for the applications we have in mind, what is more relevant is the $\varphi$-integrability of the derivative, rather than the integrability of the function which, also in view of Sobolev embeddings, could be qualified in a different way, see also Remark 24.

Celebrated Meyers-Serrin’s result was extended from the classical Sobolev spaces to the Orlicz-Sobolev space $W^1E_\varphi (\Omega )$ in [5] (see also [2]).

Theorem 18

([5]) $C^\infty (\Omega )\cap W^1E_\varphi (\Omega )$ is dense in $(W^1E_\varphi (\Omega ),\Vert \cdot \Vert _{1,\varphi })$.

It is easy to see that the previous result also fails for functions in the Orlicz-Sobolev class $W^1K_\varphi (\Omega )$, and so also in the Orlicz-Sobolev space $W^1L_\varphi (\Omega )$, provided that $(\varphi ,\Omega )$ is not $\Delta$-regular, as the following example shows.

Example 19

Assume that $n=1$, $\Omega =\,(-1,1)$, let $\varphi =\,\widetilde{\exp }_0$ be N-function in (8) with $\gamma =0$ and let

$$\begin{aligned} u(x):= \displaystyle {{\left\{ \begin{array}{ll} \frac{x}{2}\,\log \frac{1}{e|x|}&{}\text { if }|x|\le \,\frac{1}{e}\\ 0&{}\text { if }\frac{1}{e}<\,|x|\,<\,1 \end{array}\right. } } . \end{aligned}$$

Then it is easy to see that $u\in W^1K_\varphi (\Omega ){\setminus } W^1E_\varphi (\Omega )$, since the weak derivative

$$\begin{aligned} u'(x):= {\left\{ \begin{array}{ll} \log \displaystyle {\frac{1}{e\,\sqrt{|x|}}}&{}\text { if }|x|<\,\frac{1}{e}\\ 0&{}\text { if }\frac{1}{e}<\,|x|\,<\,1 \end{array}\right. } \text { a.e. }x\in \Omega \, \end{aligned}$$

belongs to $K_\varphi (\Omega )\setminus E_\varphi (\Omega )$. Indeed

$$\begin{aligned} \int _{-1}^1\varphi (|u'|)\,\,\textrm{d}x=\int _{-1}^1\left( \exp (|u'|)-|u'|-1 \right) \,\,\textrm{d}x<\,\infty , \end{aligned}$$

but $2|u'|\notin K_\varphi (\Omega )$, since

$$\begin{aligned} \int _{-1}^1\varphi (2|u'|)\,\,\textrm{d}x=\int _{-1}^1\left( \exp (2|u'|)-2|u'|-1 \right) \,\,\textrm{d}x=\,\infty . \end{aligned}$$

Thus $|u'|\notin E_\varphi (\Omega )$, since $E_\varphi (\Omega )$ is a linear subspace. By contradiction, assume there exists a sequence $(u_h)_h\subset C^\infty (\Omega )\cap W^1L_\varphi (\Omega )$ such that $u_h\rightarrow u$ in $W^1L_\varphi (\Omega )$, as $h\rightarrow \infty$. In particular, it also follows that

$$\begin{aligned} u'_h\rightarrow u' \text { in } L_\varphi (\Omega )\text { as }h\rightarrow \infty . \end{aligned}$$

(19)

Let $\psi \in C^0_c(\Omega )$ such that $0\le \,\psi \le \,1$ and $\psi \equiv \,1$ in $(-1/e, 1/e)$ and let

$$\begin{aligned} v_h:=\,\psi \,u'_h. \end{aligned}$$

By Proposition 11(iii) and (19), it still holds that

$$\begin{aligned} E_\varphi \ni \psi \,u'_h\rightarrow \psi \,u'=\,u'\text { in } L_\varphi (\Omega )\text {, as }h\rightarrow \infty . \end{aligned}$$

Then a contradiction since $u'\notin E_\varphi$.

A weaker density result of regular functions in $W^1L_\varphi (\Omega )$ holds by using the modular convergence, as shown in [6].

Theorem 20

( [6]) Let $u\in W^1L_\varphi (\Omega )$. Then there exist $\lambda >\,0$ and a sequence of functions $(u_h)_h\subset C^\infty (\Omega )\cap W^1L_\varphi (\Omega )$ such that

$$\begin{aligned} N_\varphi \left( \frac{u_h-u}{\lambda }\right) \rightarrow 0\text { and }N_\varphi \left( \frac{D_iu_h-D_iu}{\lambda }\right) \rightarrow 0\text { as }h\rightarrow \infty , \end{aligned}$$

for each $i=1,\ldots ,n$. In particular it suffices to choose $\lambda$ such that $\frac{16}{\lambda }D_iu\in K_\varphi (\Omega )$.

3 Exponential and sub-exponential N-functions

It is easy to see that a convex function $\varphi$ with polynomial growth satisfies (1), see Remark 6. We will show in Lemma 21 that $\exp _{\gamma ,\tau }^*$ satisfies the conditions in Theorem 1 for $\varphi$ with $k_\varphi =1$, if $\tau$ is sufficiently large.

Recall from (6) and (7) that we have set

$$\begin{aligned} \exp _{\gamma ,\tau }(t):=\exp \left( \displaystyle \frac{t}{(\log (t+\tau ))^\gamma }\right) , \quad \text {and}\quad \exp _{\gamma ,\tau }^*(t):=\,\exp _{\gamma ,\tau }(t)-1. \end{aligned}$$

The functions $\exp _{\gamma ,\tau }^*$, though convex, do not have null derivative at 0, and therefore do not fit exactly in the theory of N-functions. Therefore, in order to provide a bridge with the theory of N-functions of Orlicz spaces, we will also consider the modified functions

$$\begin{aligned} \widetilde{\exp }_{\gamma ,\tau }(t):=\,\exp _{\gamma ,\tau }(t)-1-\frac{t}{(\log \tau )^\gamma }= \exp _{\gamma ,\tau }^*(t)-\frac{t}{(\log \tau )^\gamma }\,, \end{aligned}$$

which are indeed N-functions and can be treated by comparison with $\exp _{\gamma ,\tau }^*$.

Lemma 21

There exists $\tau _0>0$ such that, for all $\tau \ge \tau _0$ and $0\le \gamma \le 1$, one has

• (i) $\exp _{\gamma ,\tau }$ is a smooth strictly convex increasing function. Moreover, for all $t,s\in [0,\infty )$,

  $$\begin{aligned} \exp _{\gamma ,\tau }(t+s)\le \,\exp _{\gamma ,\tau }(t)\,\exp _{\gamma ,\tau }(s), \end{aligned}$$

  (20)

  and $\exp _{\gamma ,\tau }^*$ satisfies (1) with $k_\varphi =1$, that is,

  $$\begin{aligned} \exp _{\gamma ,\tau }^*(t+s)\le \,\exp _{\gamma ,\tau }^*(t)\,\exp _{\gamma ,\tau }^*(s)+\exp _{\gamma ,\tau }^*(t)+\exp _{\gamma ,\tau }^*(s). \end{aligned}$$

  (21)

• (ii) $\widetilde{\exp }_{\gamma ,\tau }$ is a N-function satisfying

  $$\begin{aligned} \widetilde{\exp }_{\gamma ,\tau }(t)\le \,\exp _{\gamma ,\tau }^*(t)\text { if }t\ge \,0\text { and }\lim _{t\rightarrow \infty }\frac{\widetilde{\exp }_{\gamma ,\tau }(t)}{\exp _{\gamma ,\tau }^*(t)}=\,1. \end{aligned}$$

  (22)

Proof

(i) By a simple calculation, it is easy to see that, if $\tau >\,1$, $\exp _{\gamma ,\tau }$ is well-defined, $\exp _{\gamma ,\tau }\in C^\infty ([0,\infty ))$ and

$$\begin{aligned} \exp '_{\gamma ,\tau }(t)&=\,\exp \left( \frac{t}{(\log ( t+\tau ))^\gamma }\right) \frac{\log ( t+\tau )-\displaystyle {\frac{\gamma \,t}{t+\tau }}}{(\log (t+\tau ))^{\gamma +1}}, \\ \exp ''_{\gamma ,\tau }(t)&=\,\frac{\exp \left( \displaystyle {\frac{t}{(\log (t+\tau ))^\gamma }}\right) }{(\log (t+\tau ))^{2\gamma +2}}\Big [\left( \log (t+\tau )-\frac{\gamma \,t}{t+\tau }\right) ^2\\&\qquad \qquad -(\log (t+\tau ))^{\gamma +1}\frac{\gamma \,\tau + \gamma (t+\tau )}{(t+\tau )^2}+\frac{\gamma (\gamma +1)t (\log (t+\tau ))^\gamma }{(t+\tau )^2}\Big ] \end{aligned}$$

for all $t\ge 0$. Now, observe that, if $\tau >e$ and $t\ge 0$, then

$$\begin{aligned} \frac{\gamma \,t}{t+\tau }\le \,\gamma , \qquad \frac{\gamma \,\tau +\gamma (t+\tau )}{(t+\tau )^2}\le \,\frac{ 2\,\gamma }{\tau }, \text { and } \log (t+\tau )> 1\ge \gamma . \end{aligned}$$

(23)

Combining these inequalities, it follows that, if $\tau >e$,

$$\begin{aligned} \exp '_{\gamma ,\tau }(t)>0\text { for each }t\ge \,0, \end{aligned}$$

(24)

so that $\exp _{\gamma ,\tau }$ is strictly increasing on $[0,\infty )$. Let us now show that, for sufficiently large $\tau$, one has

$$\begin{aligned} \exp ''_{\gamma ,\tau }(t)>\,0\text { for each }t\ge \,0. \end{aligned}$$

(25)

By (23), for each $t\ge \,0$ and $\tau >\,e$, we obtain that

$$\begin{aligned} \begin{aligned} \left( \log (t+\tau )-\frac{\gamma \,t}{t+\tau }\right) ^2&-(\log (t+\tau ))^{\gamma +1}\frac{\gamma \,\tau + \gamma (t+\tau )}{(t+\tau )^2}+\frac{\gamma (\gamma +1)t\log (t+\tau ))^\gamma }{(t+\tau )^2}\\&\ge \,\left( \log (t+\tau )-\frac{\gamma \,t}{t+\tau }\right) ^2 -(\log (t+\tau ))^{\gamma +1}\frac{\gamma \,\tau + \gamma (t+\tau )}{(t+\tau )^2}\\&\ge \,\left( \log (t+\tau )-\gamma \right) ^2-\frac{2\gamma }{\tau }(\log (t+\tau ))^{2}\\&=\log (t+\tau ) \left( \log (\tau ) -2\gamma \left( \frac{\log (\tau )}{\tau }+1\right) \right) + \gamma ^2 \\&\ge \log (t+\tau ) \left( \log (\tau ) -2 \left( \frac{\log (\tau )}{\tau }+1\right) \right) . \end{aligned} \end{aligned}$$

It is clear that, there is $\tau _0>0$ (independent on $\gamma$), so that the latter quantity is positive for all $\tau \ge \tau _0$ and $t\ge 0$. Hence, (25) follows and $\exp _{\gamma ,\tau }$ is strictly convex on $[0,\infty )$.

Let us show (20), that is, for every $t,s\in [0,\infty )$,

$$\begin{aligned} \begin{aligned} \exp _{\gamma ,\tau }(t+s)&=\,\exp \left( \frac{t+s}{(\log (t+s+\tau ))^\gamma }\right) \\&\le \, \exp _{\gamma ,\tau }(t)\,\exp _{\gamma ,\tau }(s) \\&=\,\exp \left( \frac{t}{(\log (t+\tau ))^\gamma }+\frac{s}{(\log (s+\tau ))^\gamma }\right) . \end{aligned} \end{aligned}$$

(26)

Observe that

$$\begin{aligned} \begin{aligned} \frac{t+s}{(\log (t+s+\tau ))^\gamma }&=\,\frac{t}{(\log (t+s+\tau ))^\gamma }+\frac{s}{(\log (t+s+\tau ))^\gamma }\\&\le \,\frac{t}{(\log (t+\tau ))^\gamma }+\frac{s}{(\log (s+\tau ))^\gamma }, \end{aligned} \end{aligned}$$

whence (26) follows, being the exponential function nondecreasing. Inequality (21) follows by using (20) and the fact that $\exp _{\gamma ,\tau }(t)=\, \exp _{\gamma ,\tau }^*(t)+1$.

(ii) Notice that, if

$$\begin{aligned} a(t):=\,\exp '_\gamma (t)-\exp '_\gamma (0)\text { if } t\ge \,0, \end{aligned}$$

by (24) and (25), a is continuous, (strictly) increasing, $a(0)=\,0$, $a(t)>\,0$ if $t>\,0$ and $\lim _{t\rightarrow \infty }a(t)=\,\infty$. Moreover, being a increasing, we have for $t\ge 0$,

$$\begin{aligned} \widetilde{\exp }_{\gamma ,\tau }(t)=\,\exp _{\gamma ,\tau }(t)-1-\frac{t}{(\log \tau )^\gamma }=\,\int _0^t\left( \exp '_\gamma (s)-\exp '_\gamma (0)\right) \,ds=\,\int _0^ta(s)\,ds. \end{aligned}$$

Thus, $\widetilde{\exp }_{\gamma ,\tau }$ is (strictly) convex.

Finally, since the functions have a more than linear growth, it is clear that $\widetilde{\exp }_{\gamma ,\tau }(t)/\exp _{\gamma ,\tau }(t)$ tends to 1 as $t\rightarrow \infty$. $\square$

Remark 22

Notice that, by (22), the pair $(\widetilde{\exp }_{\gamma ,\tau },\Omega )$ is never $\Delta$-regular for any $\Omega \subset \mathbb {R}^n$. In particular, by Lemma 8, $K_\varphi (\Omega )$ is never a vector space if $\varphi = \widetilde{\exp }_{\gamma ,\tau }$.

Remark 23

Exponential growth functions fall also in the class treated by Theorem 1. More precisely, the functions

$$\begin{aligned} e_\alpha ^*(t):= \exp (\alpha t) - 1, \end{aligned}$$

for $\alpha >0$ satisfy the conditions on $\varphi$ given in Theorem 1.

4 Proof of the approximation results

In this section we are going to show our results.

Proof of Theorem 1

Uniform positivity of w on compact subsets gives

$$\begin{aligned} \,\int _R\varphi \left( |Db(s,x)|\right) \,\textrm{d}s \,\textrm{d}x< \infty \qquad \text {whenever }R\Subset I\times \Omega . \end{aligned}$$

(27)

We are going to exploit an adaptation of the technique of the proof of Meyers-Serrin’s theorem (see, for instance, [3, Thm. 3.9]). Let $Q:=\,I\times \Omega$ and let $U_j$, $j=0,1,\ldots$, be the nondecreasing sequence of open subsets

$$\begin{aligned} U_0:=\,\emptyset ,\quad U_j:=\,\left\{ (s,x)\in Q:\,\textrm{dist}((s,x),\partial Q)>\frac{1}{j},\, |s|+|x| < j \right\} \,\,\,(j=1,2,\ldots ), \end{aligned}$$

and let

$$\begin{aligned} Q_j:=\,U_{j+1}\setminus {\overline{U}}_{j-1}\quad j=1,2,\ldots . \end{aligned}$$

Then $\cup _jQ_j=Q$, each $Q_j$ has compact closure in Q and any point of Q belongs to at most four sets $Q_j$. More specifically, if $j\ge \,3$ and $x\in Q_j$, then x may belong at most to $Q_{j-1}$ and $Q_{j+1}$.

Let $(\zeta _j)_j$ be a partition of unity relative to the covering $(Q_j)$, that is, nonnegative functions $\zeta _j \in C^\infty _c(Q_j)$ such that $\sum _{j=1}^\infty \zeta _j\equiv \,1$ in Q. Moreover, let $\psi _j\in C^\infty _c(Q)$ be cut-off functions such that $0\le \,\psi _j\le \,1$ in Q and $\psi _j\equiv \,1$ in $Q_j$.

For each $j=1,2,\ldots$, let $b_j:\,\mathbb {R}^{n+1}=\,\mathbb {R}_s\times \mathbb {R}^n_x\rightarrow \mathbb {R}^m$ denote

$$\begin{aligned} b_j(s,x):= {\left\{ \begin{array}{ll} \,\psi _j(s,x)\,b(s,x) &{}\text { if }(s,x)\in I\times \Omega \\ 0&{}\text { if }(s,x)\in \mathbb {R}^{n+1}\setminus I\times \Omega , \end{array}\right. } \end{aligned}$$

so that it is clear that $\textrm{spt}(b_j)\Subset Q$, $b_j\in L^1(\mathbb {R}_s;W^{1,1}(\mathbb {R}^n_x;\mathbb {R}^m))\cap C^0_\textrm{c}(\mathbb {R}^{n+1},\mathbb {R}^m)$, and

$$\begin{aligned} Db_j=\,D(\psi _j\,b)=\,\psi _j\,D b+\nabla \psi _j\otimes b\text { a.e.~in }Q. \end{aligned}$$

(28)

In particular,

$$\begin{aligned} b_j=\,b\text { and } Db_j=\,D b\text { a.e.~in }Q_j\,, \end{aligned}$$

(29)

$$\begin{aligned} Db_j\in L^1(\mathbb {R}^{n+1};\mathbb {R}^{nm})\,. \end{aligned}$$

(30)

The monotonicity of $\varphi$ and the weak subadditivity condition (1) give

$$\begin{aligned} \begin{aligned} \varphi \left( |Db_j|\right)&\le \,\varphi \left( \psi _j\,|D b|+|\nabla \psi _j\otimes b|\right) \\&\le \,k_\varphi \left[ \varphi \left( \psi _j\,|D b|\right) \,\varphi \left( |\nabla \psi _j\otimes b|\right) +\varphi \left( \psi _j\,|D b|\right) +\varphi \left( |\nabla \psi _j\otimes b|\right) \right] . \end{aligned} \end{aligned}$$

(31)

Since, by (27),

$$\begin{aligned} \varphi \left( \psi _j\,|D b|\right) \in L^1(\mathbb {R}^{n+1})\quad \text { and } \quad \varphi \left( |\nabla \psi _j\otimes b|\right) \in L^1(\mathbb {R}^{n+1})\cap L^\infty (\mathbb {R}^{n+1}), \end{aligned}$$

we obtain from (31) that

$$\begin{aligned} \varphi \left( |Db_j|\right) \in L^1(\mathbb {R}^{n+1}). \end{aligned}$$

(32)

Let $(\rho _\varepsilon (s,x))_\varepsilon$ be space-time mollifiers on $\mathbb {R}^{n+1}=\,\mathbb {R}_s\times \mathbb {R}^n_x$. For each $\delta \in (0,1)$ we will make a suitable choice of $0<\varepsilon _j<\delta$ and define $b_\delta :\,Q\rightarrow \mathbb {R}^m$ as

$$\begin{aligned} b_\delta (s,x):= \sum _{j=1}^\infty \zeta _j(s,x)\,(\rho _{\epsilon _j}*b_j)(s,x). \end{aligned}$$

Since the sum is locally finite, $b_\delta$ is well defined. Moreover, by construction, $b_\delta \in C^\infty (I\times \Omega ;\mathbb {R}^m)$ and one has

$$\begin{aligned} b_\delta \rightarrow b\text { in }L^1_\textrm{loc}(Q;\mathbb {R}^m)\text {, as } \delta \rightarrow 0. \end{aligned}$$

(33)

Notice now that

$$\begin{aligned} \begin{aligned} Db_\delta&=\,\sum _{j=1}^\infty D(\zeta _j(\rho _{\varepsilon _j}*b_j))=\,\sum _{j=1}^\infty \zeta _j(\rho _{\varepsilon _j}*Db_j)+\sum _{j=1}^\infty \nabla \zeta _j\otimes (\rho _{\varepsilon _j}* b)\\&=\,v_\delta +z_\delta \text { in }Q, \end{aligned} \end{aligned}$$

(34)

where

$$\begin{aligned} v_\delta :=\sum _{j=1}^\infty \zeta _j(\rho _{\varepsilon _j}*Db_j), \end{aligned}$$

and

$$\begin{aligned} z_\delta :=\sum _{j=1}^\infty \left( \nabla \zeta _j\otimes (\rho _{\varepsilon _j}* b_j)-\nabla \zeta _j\otimes b_j\right) , \end{aligned}$$

where we used the fact that, since $\sum _{j=1}^\infty \nabla \zeta _j\equiv 0$, we have $\sum _{j=1}^\infty \nabla \zeta _j\otimes b\equiv 0$.

For each $\delta >\,0$ and $j=1,2,\ldots$, we can find $0<\varepsilon _j<\delta$ such that

$$\begin{aligned} \int _Q|\zeta _j\,(\rho _{\varepsilon _j}*b_j)-\zeta _j\,b|\,\textrm{d}x\,\textrm{d}s<\frac{\delta }{2^j}\, \end{aligned}$$

(35)

and

$$\begin{aligned} \Vert \zeta _j(\rho _{\varepsilon _j}*Db_j)-\zeta _jDb_j\Vert _{L^1(\mathbb {R}^{n+1};\mathbb {R}^m)} <\frac{\delta }{2^{j+1}}. \end{aligned}$$

(36)

In addition, setting

$$\begin{aligned} M_j:=\,\max \{1,\sup _{Q_j}w\} \,, \end{aligned}$$

we can also ensure that

$$\begin{aligned} \begin{aligned}&\Vert \nabla \zeta _j\otimes (\rho _{\varepsilon _j}* b_j)-\nabla \zeta _j\otimes b\Vert _{L^p(Q_j;\mathbb {R}^{nm})}\\&=\,\Vert \nabla \zeta _j\otimes (\rho _{\varepsilon _j}* b_j)-\nabla \zeta _j\otimes b\Vert _{L^p(\mathbb {R}^{n+1};\mathbb {R}^{nm})}<\frac{\delta }{2^{j+1} M_j}\text { if }p=1,\infty \, \end{aligned} \end{aligned}$$

(37)

and

$$\begin{aligned} \Vert \rho _{\varepsilon _j}*\varphi (|Db_j|)-\varphi (|Db_j|)\Vert _{L^1(\mathbb {R}^{n+1})} <\frac{\delta }{2^j\,M_j}. \end{aligned}$$

(38)

Notice that, by assumption on w, $M_j<\infty$. Notice also that the choices in (35) and (37) are possible thanks to the continuity of b (in particular, for (37) with $p=\infty$, we are using also continuity of b with respect to the time variable), while the choices in (36) and (38) are possible thanks to the classical properties of convolution together with (30) and (32), respectively. From (37), it follows that

$$\begin{aligned} \Vert z_\delta \Vert _{L^p(Q;\mathbb {R}^{nm})}< \frac{\delta }{2}\text { if }p=1,\infty , \text { and}\int _Q |z_\delta | \,w\,\textrm{d}x\,\textrm{d}s <\delta . \end{aligned}$$

(39)

Moreover, by (39) with $p=\,\infty$ and the convexity of $\varphi$, if we set $L:=\varphi (1)$, then $\varphi (0)=0$ and the monotonicity of difference quotients give

$$\begin{aligned} \sigma _\delta :=\varphi (|z_\delta |)\le \,L |z_\delta |\text { in }Q. \end{aligned}$$

(40)

Notice now that, since $Db=\,\sum _{j=1}^\infty \zeta _j Db=\,\sum _{j=1}^\infty \zeta _j Db_j$, by (36) and (39) with $p=1$, we have

$$\begin{aligned} \begin{aligned} \Vert Db_\delta -Db\Vert _{L^1(Q;\mathbb {R}^m)}&=\,\Vert v_\delta +z_\delta -Db\Vert _{L^1(Q;\mathbb {R}^m)}\\&\le \,\Vert v_\delta -Db\Vert _{L^1(Q;\mathbb {R}^m)}+\,\Vert z_\delta \Vert _{L^1(Q;\mathbb {R}^m)}\\&\le \,\sum _{j=1}^\infty \Vert \zeta _j(\rho _{\varepsilon _j}*Db_j)-\zeta _jDb_j\Vert _{L^1(\mathbb {R}^{n+1};\mathbb {R}^m)}+\frac{\delta }{2}\\&\le \, \sum _{j=1}^\infty \frac{\delta }{2^{j+1}}+\frac{\delta }{2}=\,\delta . \end{aligned} \end{aligned}$$

(41)

By (41), it follows that

$$\begin{aligned} \lim _{\delta \rightarrow 0} \Vert Db_\delta -Db\Vert _{L^1(Q;\mathbb {R}^m)} = 0. \end{aligned}$$

(42)

In particular, by the continuity of $\varphi$ and (42), there exists an infinitesimal sequence $(\delta _h)_h$ such that, if $b_h:=\,b_{\delta _h}$,

$$\begin{aligned} \varphi \left( |Db_h|\right) \rightarrow \varphi \left( |Db|\right) \text {a.e. in }Q\text {, as }h\rightarrow \infty . \end{aligned}$$

(43)

Let us now show (4). One has, a.e. in Q,

$$\begin{aligned} \begin{aligned} \varphi \left( |Db_{\delta }|\right)&\le \varphi \left( |z_\delta |+ |v_\delta |\right) \le \,k_\varphi \left( \varphi \left( |z_\delta |\right) \,\varphi \left( |v_\delta |\right) +\varphi \left( |z_\delta |\right) +\varphi \left( |v_\delta |\right) \right) \\&\le \, k_\varphi ( (1+\sigma _\delta )\,\varphi \left( \left| v_\delta \right| \right) +\sigma _\delta ), \end{aligned} \end{aligned}$$

where $\sigma _\delta =\varphi \left( |z_\delta |\right)$. Set $\sigma _\delta ^\infty := \Vert \sigma _\delta \Vert _{L^\infty (Q)}$, so that, a.e. in Q,

$$\begin{aligned} \varphi \left( |Db_\delta |\right) \le \, k_\varphi ((1+\sigma _\delta ^\infty )\,\varphi \left( \left| v_\delta \right| \right) +\sigma _\delta ). \end{aligned}$$

(44)

By the monotonicity and convexity of $\varphi$, by Jensen’s inequality and taking into account that $(\zeta _j)_j$ is a partition of unity, we get, a.e. in Q,

$$\begin{aligned} \begin{aligned} \varphi \left( \left| v_\delta \right| \right)&=\,\varphi \left( \left| \sum _{j=1}^\infty \zeta _j(\rho _{\varepsilon _j}*Db_j)\right| \right) \le \,\varphi \left( \sum _{j=1}^\infty \zeta _j (\rho _{\varepsilon _j}* |Db_j|)\right) \\&\le \,\sum _{j=1}^\infty \zeta _j (\rho _{\varepsilon _j}*\varphi \left( |Db_j|\right) )=:\, G_\delta . \end{aligned} \end{aligned}$$

(45)

Hence, by (44) and (45), if follows that

$$\begin{aligned} w\varphi \left( |Db_\delta |\right) \le \,k_\varphi ((1+\sigma _\delta ^\infty )\,wG_\delta +\, w\sigma _\delta ) \quad \text {a.e. in }Q. \end{aligned}$$

(46)

It is clear that, by (39) and (40),

$$\begin{aligned} w\,\sigma _\delta \text { converges to }0\text { in }L^1(Q)\text { and } \sigma _\delta ^\infty \rightarrow 0\text {, as }\delta \rightarrow 0. \end{aligned}$$

(47)

Let us now prove that

$$\begin{aligned} w\,G_{\delta } \text { converges to}\, w\varphi (|Db|) \text { in } L^1(Q), \text {as } \delta \rightarrow 0. \end{aligned}$$

(48)

Observe that, by (29),

$$\begin{aligned} \varphi (|Db|)=\,\sum _{j=1}^\infty \zeta _j\varphi (|Db|)=\,\sum _{j=1}^\infty \zeta _j\varphi (|Db_j|)\text { a.e. in }Q, \end{aligned}$$

so that (38) gives (48).

The combination of (48) and (46) gives the equi-integrability of $w\varphi \left( |Db_\delta |\right)$. By using (43), Vitali’s form of the dominated convergence theorem (see, for instance, [3, Exercise 1.18]), gives (4) and the proof is complete. $\square$

Remark 24

Let $\Psi :[0,\infty )\rightarrow [0,\infty )$ be any continuous function with $\Psi (0)=0$ and linear growth at the origin. Notice that all the terms $\zeta _j(\rho _{\epsilon _j}*b_j)-\zeta _jb$ can be made arbitrarily small not only in $L^\infty _t(L^\infty _x)$, but also in $L^1_t(L^1_x)$, choosing $\epsilon _j\ll 1$. Then, using the representation

$$\begin{aligned} (b_{\delta _h}-b)=\sum _{j=1}^\infty \zeta _j(\rho _{\epsilon _j}*b_j)-\zeta _jb \end{aligned}$$

we can improve the construction to get also

$$\begin{aligned} \lim _{h\rightarrow \infty }\int _I\int _\Omega \Psi (|b_{\delta _h}-b|)\,\textrm{d}x\,\textrm{d}s=0. \end{aligned}$$

More precisely, choosing $\epsilon _j\ll 1$ properly, we can make arbitrarily small all terms

$$\begin{aligned} \int _{Q_j} \Psi (|\sum _{j=1}^\infty \zeta _j(\rho _{\epsilon _j}*b_j)-\zeta _jb|)\,\textrm{d}x\,\textrm{d}s \end{aligned}$$

since the sum is locally finite and $Q_j\Subset I\times \Omega$.

Remark 25

Since the proof of Theorem 1 is based on a convolution argument, we have more control on the convergence depending on the properties of b. We give two cases that can be of interest.

First, if $b\in L^1(I;C(\Omega ;\mathbb {R}^m))$ is continuous in the spatial variable, then the approximating sequence $b_h\in C^\infty (I\times \Omega ;\mathbb {R}^m)$ in Theorem 1 can be taken so that $b_h\rightarrow b$ in $b\in L^1(I;C(\Omega ;\mathbb {R}^m))$.

Second, if there exists a bounded open set $\Omega '\Subset \Omega$ such that

$$\begin{aligned} \textrm{spt}(b(t,\cdot ))\subset \Omega '\text { for every }t\in I, \end{aligned}$$

(49)

then the approximating sequence $b_h\in C^\infty (I\times \Omega ;\mathbb {R}^m)$ can be taken with

$$\begin{aligned} \textrm{spt}(b_h(t,\cdot ))\subset \Omega ' \text { for each }t\in I \text { and }h\in \mathbb {N}. \end{aligned}$$

(50)

Proof of Corollary 2

Let us prove that

$$\begin{aligned} \int _I\int _\Omega w(s,x)\exp _{\gamma ,\tau }^*\left( |Db(s,x)|\right) \,\textrm{d}x \,\textrm{d}s<\infty . \end{aligned}$$

(51)

Notice that, since b satisfies (2) with (9), then $w\,\widetilde{\exp }_{\gamma ,\tau }\left( |Db|\right) \in L^1(I\times \Omega )$. On the one hand, if $w|Db|\in L^1(I\times \Omega )$, since

$$\begin{aligned} w\exp _{\gamma ,\tau }^*\left( |Db|\right) =\,w\widetilde{\exp }_{\gamma ,\tau }\left( |Db|\right) +\frac{w|Db|}{(\log \tau )^\gamma } \end{aligned}$$

and $w\,\widetilde{\exp }_{\gamma ,\tau }\left( |Db|\right) \in L^1(I\times \Omega )$ we immediately obtain (51). On the other hand, if $w\in L^1(I\times \Omega )$, by (22), there exists ${{\bar{t}}}>\,0$ such that

$$\begin{aligned} \frac{1}{2} \, \exp _{\gamma ,\tau }^*(t)\le \, \widetilde{\exp }_{\gamma ,\tau }(t)\text { for each }t\ge \,{{\bar{t}}}. \end{aligned}$$

(52)

Thus

$$\begin{aligned} \begin{aligned} \int _{I\times \Omega }w\,\exp _{\gamma ,\tau }^*\left( |Db|\right) \,\,\textrm{d}s \,\textrm{d}x&=\,\int _{\{|Db|<\,{{\bar{t}}}\}}w\,\exp _{\gamma ,\tau }^*\left( |Db|\right) \,\,\textrm{d}s \,\textrm{d}x \\&\qquad \qquad +\int _{\{|Db|\ge \,{{\bar{t}}}\}}w\,\exp _{\gamma ,\tau }^*\left( |Db|\right) \,\,\textrm{d}s \,\textrm{d}x\\&\le \,\exp _{\gamma ,\tau }^*\left( {{\bar{t}}}\right) \int _{I\times \Omega }w\,\,\textrm{d}s \,\textrm{d}x\\&\qquad \qquad +2\int _{I\times \Omega }w\,\widetilde{\exp }_{\gamma ,\tau }\left( |Db|\right) \,\,\textrm{d}s \,\textrm{d}x<\,\infty \end{aligned} \end{aligned}$$

and (51) follows once more.

By (51), we can apply Theorem 1 to get the existence of $b_h\in C^\infty (I\times \Omega ;\mathbb {R}^m)$ satisfying (3) and (4) with $\varphi =\,\exp _{\gamma ,\tau }^*$. Since $w\,\widetilde{\exp }_{\gamma ,\tau }\left( |Db_h|\right) \le \,w\,\exp _{\gamma ,\tau }^*\left( |Db_h|\right)$, by applying Vitali’s convergence theorem, we obtain again the desired conclusion. $\square$

In the proof of Theorem 3 we will need the following lemma.

Lemma 26

Let $f\in L^1_\textrm{loc}(\mathbb {R}_s\times \Omega )$ and $(\rho _\varepsilon (s))_\varepsilon$ be a family of time mollifiers in $\mathbb {R}_s$. Then, for a.e. $x\in \Omega$, the time convolution product $f^\varepsilon (\cdot ,x):\mathbb {R}_s\rightarrow \mathbb {R}$

$$\begin{aligned} \begin{aligned} f^\varepsilon (s,x)&=\,(\rho _\varepsilon *f(\cdot ,x))(s)\\&:=\int _{\mathbb {R}}\rho _\varepsilon (s-v)\,f(v,x)\,dv\text { for each }s\in \mathbb {R}\end{aligned} \end{aligned}$$

and

$$\begin{aligned} f^\varepsilon (\cdot ,x)\in C^0(\mathbb {R}_s)\text { for each } \varepsilon >\,0, \text { for a.e. }x\in \Omega . \end{aligned}$$

(53)

In addition, for any open set $\omega \subset \Omega$ one has

$$\begin{aligned} \Vert f^\varepsilon \Vert _{L^1(\mathbb {R}_s\times \omega )}\le \,\Vert f\Vert _{L^1(\mathbb {R}_s\times \omega )}\text { for each } \varepsilon >\,0 \end{aligned}$$

(54)

and

$$\begin{aligned} f^\varepsilon \rightarrow f\text { in }L^1(\mathbb {R}_s\times \omega )\text { as }\varepsilon \rightarrow 0, \end{aligned}$$

(55)

provided that $f\in L^1(\mathbb {R}_s\times \omega )$.

Finally, if we assume that, for each ball $B(x_0,r)\Subset \Omega$ one has

$$\begin{aligned} f\in L^1\bigl (\mathbb {R}_s;C^0(B(x_0,r))\bigr ), \end{aligned}$$

(56)

then

$$\begin{aligned} f^\varepsilon \in C^0(\mathbb {R}_s\times \Omega )\text { for each }\varepsilon >\,0. \end{aligned}$$

(57)

Proof

Properties (53), (54) and (55) can be proved as in the case of the global (s, x)-convolution by mollifiers (see, for instance, [3, Section 2.1]). Let us prove (57). Let $(s_0,x_0)\in \mathbb {R}_s\times \Omega$ and let $((s_h,x_h))_h\subset \mathbb {R}_s\times \Omega$ a sequence converging to $(s_0,x_0)$. From (56),

$$\begin{aligned} \int _\mathbb {R}F(s)\,ds<\infty \text { if }F(s):=\,\sup _{B(x_0,r)}|f(s,\cdot )|, \end{aligned}$$

and, without loss of generality, we can assume that $(x_h)_h\subset B(x_0,r)$ for a fixed $r>0$. Then, since

$$\begin{aligned} |\rho _\varepsilon (s_h-v)\,f(v,x_h)|\le \,\sup _\mathbb {R}\rho _\varepsilon \,F(v)\quad \text {for a.e.} v\in \mathbb {R}, \end{aligned}$$

by Lebesgue’s dominated convergence theorem, it follows that

$$\begin{aligned} \begin{aligned} \lim _{h\rightarrow \infty }f^\varepsilon (s_h,x_h)&=\,\lim _{h\rightarrow \infty }\int _\mathbb {R}\rho _\varepsilon (s_h-v)\,f(v,x_h)\,dv\\&=\,\int _\mathbb {R}\lim _{h\rightarrow \infty }( \rho _\varepsilon (s_h-v)\,f(v,x_h))\,dv\\&=\,\int _\mathbb {R}\rho _\varepsilon (s_0-v)\,f(v,x_0)\,dv=\,f^\varepsilon (s_0,x_0). \end{aligned} \end{aligned}$$

$\square$

Proof of Theorem 3

We extend b to $\mathbb {R}\times \Omega$ setting $b(t,x)=0$ whenever $t\notin I$. Denoting by $b^\epsilon$ the mollified functions with respect to the time variable, one has

$$\begin{aligned} Db^\epsilon (t,x)=\int _\mathbb {R}\rho _\epsilon (t-s)Db(s)\,\textrm{d}s \end{aligned}$$

and therefore Jensen’s inequality gives

$$\begin{aligned} \int _\Omega w(x)\varphi (Db^\epsilon (t,x))\,\textrm{d}x\le & {} \int _\Omega w(x)\int _\mathbb {R}\rho _\epsilon (t-s)\varphi (|Db(s,x)|)\,\textrm{d}s\,\textrm{d}x\\= & {} \int _\mathbb {R}\rho _\epsilon (t-s)\int _\Omega w(x)\varphi (Db(s,x)|)\,\textrm{d}x\,\textrm{d}s. \end{aligned}$$

By integration on I, it follows that $N_{\varphi ,w}(|Db^\epsilon |)\le N_{\varphi ,w}(|Db|)$. Notice now that $b^\epsilon$ satisfy the assumptions of Corollary 2, thanks to (57). Thus for all $\epsilon >0$ we get the existence of a sequence $(b^\epsilon _h)_h\subset C^\infty (I\times \Omega ;\mathbb {R}^m)$ satisfying (3) and (4). Finally, by taking a diagonal sequence, we get the desired conclusion. $\square$

Proof of Theorem 4

Notice that for any open set $A\subset \Omega$ the weak $L^1(A;\mathbb {R}^{mn})$ convergence of derivatives grants

$$\begin{aligned} \liminf _{h\rightarrow \infty }\int _A \varphi \left( |Db_h|\right) \,\textrm{d}x \ge \int _A \varphi \left( |Db|\right) \,\textrm{d}x\,. \end{aligned}$$

Therefore, by applying an elementary lemma (see, for instance, the proof of Proposition 1.80 in [3]) the convergence of the integrals on $\Omega$ can be localized, getting

$$\begin{aligned} \lim _{h\rightarrow \infty }\int _A \varphi \left( |Db_h|\right) \,\textrm{d}x =\int _A \varphi \left( |Db|\right) \,\textrm{d}x \end{aligned}$$

whenever $A\subset \Omega$ is open with Lebesgue negligible boundary. In particular, choosing $A\Subset \Omega$ with this property, since A has finite measure we can use the strict convexity of $\varphi$ and [11] to get that $Db_h\rightarrow Db$ in $L^1(A;\mathbb {R}^{mn})$. It follows that $Db_h\rightarrow Db$ in $L^1_\textrm{loc}(I\times \Omega ;\mathbb {R}^{nm})$ and therefore, modulo the extraction of a subsequence, we can assume that $Db_h\rightarrow Db$ a.e. in $\Omega$.

Combining the pointwise convergence

$$\begin{aligned} \lim _{h\rightarrow \infty }\varphi (|Db_h|)=\varphi (|Db|)\quad \text {a.e. in } \Omega \end{aligned}$$

with the convergence of the integrals, Scheffé’s lemma gives that $\varphi (|Db_h|)$ converges in $L^1(\Omega )$ to $\varphi (|Db|)$. Now, the inequality

$$\begin{aligned} \varphi \left( \frac{|Db_h-Db|}{2}\right) \le \frac{1}{2} \varphi \left( |Db_h|\right) +\frac{1}{2} \varphi \left( |Db|\right) \end{aligned}$$

grants the equi-integrability of $\varphi (|Db_h-Db|/2)$. Vitali’s convergence theorem can finally be applied to get the result. $\square$

Proof of Theorem 5

Notice that, being $\widetilde{\exp }$ a N-function, it has a linear growth at the origin. Thus by applying Remark 24 with $\Psi =\widetilde{\exp }$ and Corollary  2 with $b(t,x)=\,u(x)$ and $w\equiv 1$, we get the desired conclusion. $\square$