Let $G\subset {\mathbb{R}}^d$ be an open set with Lipschitz boundary. Let A be a Young function satisfying the ${\mathrm{\Delta }}_2$-condition. Then $C^{\infty }(\bar{G})$ is dense in $H^A(G)$.

We follow the lines of the proof in [137, Thm. 1.1] and split the proof into three parts. Let $\mathbf{u}\in H^A(G)$ be given.

i) We show that u can be approximated by functions with compact support. Take a function ${\phi }_n\in C_0^{\infty }(B_{2n}(0))$ with $0⩽\phi ⩽1$ and $\phi =1$ in $B_n(0)$. It is easy to check that ${\phi }_n\mathbf{u}{|}_G$ converges to u in $H^A(G)$.

ii) Let $G={\mathbb{R}}^d$, so $\mathbf{u}\in H^A({\mathbb{R}}^d)$ and by i) we can assume that u has compact support. Let ϱ be a standard mollifier, i.e. $\varrho \in C_0^{\infty }(B_1(0))$, $\varrho ⩾0$ and $\int \varrho \mathrm{d}x=1$. We set ${\varrho }_{\epsilon }(x)={\epsilon }^{-d}\varrho \left(\frac{x}{\epsilon }\right)$ and ${\mathbf{u}}_{\epsilon }=\mathbf{u}⁎{\varrho }_{\epsilon }$. Of course this means that ${\mathbf{u}}_{\epsilon }\in C_0^{\infty }({\mathbb{R}}^d)$. Moreover, due to ${\mathrm{\Delta }}_2$-condition of A we have for $\epsilon \to 0$

${\mathbf{u}}_{\epsilon }⟶\mathbf{u}\text{in}{L}^A({\mathbb{R}}^d).$

As $\mathrm{div}{\mathbf{u}}_{\epsilon }=\mathrm{div}(\mathbf{u}⁎{\varrho }_{\epsilon })=(\mathrm{div}\mathbf{u})⁎{\varrho }_{\epsilon }$ we have

$\mathrm{div}{\mathbf{u}}_{\epsilon }⟶\mathrm{div}\mathbf{u}\text{in}{L}^A({\mathbb{R}}^d)$

as well. Both together yields that ${\mathbf{u}}_{\epsilon }$ converges to u in $H^A(G)$.

iii) For the general case $G\ne {\mathbb{R}}^d$ we can assume that G is locally star-shaped. There is an open covering $G,{({\mathcal{O}}_j)}_{j\in J}$ of $\bar{G}$. Let $\phi ,{({\phi }_j)}_{j\in J}$ be a partition of unity subordinated to this covering, where $\phi \in C_0^{\infty }(G)$ and ${\phi }_j\in C_0^{\infty }({\mathcal{O}}_j)$. We have

$\mathbf{u}=\phi \mathbf{u}+\sum _{j\in J}{\phi }_j\mathbf{u}.$

The sum is finite as we can assume that u has compact support by i). The function $\phi \mathbf{u}$ has compact support in G and by standard mollification as in ii) it can be approximated by a sequence in $C_0^{\infty }(G)$. So it remains to show that we can approximate the functions ${\phi }_j\mathbf{u}$. The set ${\mathcal{O}}_j^{\prime }:={\mathcal{O}}_j\cap G$ is star-shaped with respect to some of its points; for simplicity we assume that this point is 0. We consider the function ${\mathbf{u}}_j={\phi }_j\mathbf{u}$ and for ${\mathbf{u}}_j^{\lambda }={\mathbf{u}}_j(\lambda \cdot )$ for $\lambda >1$. It is easy to show that ${\mathbf{u}}_j^{\lambda }{|}_{{\mathcal{O}}_j^{\prime }}$ converges to ${\mathbf{u}}_j$ in $H^A({\mathcal{O}}_j^{\prime })$ for $\lambda \to 1$. So it is enough to approximate ${\mathbf{u}}_j^{\lambda }$ instead of ${\mathbf{u}}_j$. The function ${\mathbf{u}}_j^{\lambda }$ as compact support and hence belongs to $H^A({\mathbb{R}}^d)$. On account of ii) it can be approximated in $H^A({\mathbb{R}}^d)$ by functions from $C_0^{\infty }({\mathbb{R}}^d)\subset C^{\infty }(\bar{G})$. Restricting to G yields the convergence in $H^A(G)$ we are looking for.

 □

Let $G\subset {\mathbb{R}}^d$ be an open set with Lipschitz boundary. Let A be a Young function satisfying the ${\mathrm{\Delta }}_2$-condition. Then there holds

$H_0^A(G)\subset {\overline{C_0^{\infty }(G)}}^{{\Vert \cdot \Vert }_{H^A(G)}}.$

Let $\tilde{\mathbf{u}}$ be the extension of u to ${\mathbb{R}}^d$ by 0. By (A.1.2) we have

${\int }_{{\mathbb{R}}^d}\tilde{\mathbf{u}}\cdot \mathrm{\nabla }\phi \mathrm{d}x+{\int }_{{\mathbb{R}}^d}\mathrm{div}\mathbf{u}\phi \mathrm{d}x=0$

for every $\phi \in C_0^{\infty }({\mathbb{R}}^d)$. Hence $\mathrm{div}\tilde{\mathbf{u}}={\chi }_G\mathrm{div}\mathbf{u}$ and so

${\int }_{{\mathbb{R}}^d}\tilde{\mathbf{u}}\cdot \mathrm{\nabla }\phi \mathrm{d}x+{\int }_{{\mathbb{R}}^d}\mathrm{div}\tilde{\mathbf{u}}\phi \mathrm{d}x=0$

and $\tilde{\mathbf{u}}\in H^A(G)$. We now follow the ideas from the proof of Theorem A.1.45 and can reduce the situation to the case that G is star-shaped with respect to 0. For $\lambda >1$ we consider ${\tilde{\mathbf{u}}}_{\lambda }=\tilde{\mathbf{u}}(\lambda \cdot )$ and have ${\tilde{\mathbf{u}}}_{\lambda }\to \tilde{\mathbf{u}}$ in $H^A(G)$ for $\lambda \to 1$. But ${\tilde{\mathbf{u}}}_{\lambda }$ has compact support in G and so has the mollification ${({\tilde{\mathbf{u}}}_{\lambda })}_{\epsilon }$ for ε small enough. On account of the properties of the mollification (using the ${\mathrm{\Delta }}_2$-condition) we are able to approximate u by a sequence of $C_0^{\infty }(G)$-functions.  □

It is possible to show ${\overline{C_0^{\infty }(G)}}^{{\Vert \cdot \Vert }_{H^A(G)}}\subset H_0^A(G)$ provided the boundary of G is smooth enough. In fact one has to introduce the trace ${\gamma }_{\mathcal{N}}(\mathbf{u})=\mathcal{N}\cdot \mathbf{u}{|}_{\partial G}$ in a negative Sobolev space and follow the ideas in [137, Thm. 1.3]. In the case studied there the surjectivity of the trace map $W^{1,2}(G)\to W^{1/2,2}(\partial G)$ is used. An analogone to this in Orlicz spaces is not known. So a Lipschitz boundary might not be sufficient.

a) The space $\mathrm{BD}(G)$ is continuously embedded into the Lebesgue space $L^{d/(d-1)}(G)$.

b) For $1\le p<d/(d-1)$ the embedding $\mathrm{BD}(G)\hookrightarrow L^p(G)$ is compact.

a) There is continuous linear operator

$\gamma :\mathrm{BD}(G)\to L^1(\partial G,{\mathcal{H}}^{d-1})$

with $\gamma (\mathbf{u})=\mathbf{u}{|}_{\partial G}$ for all $\mathbf{u}\in C(\bar{G})$.

b) Let $\mathbf{u},{\mathbf{u}}_k\in \mathrm{BD}(G)$ with ${\mathbf{u}}_k\to \mathbf{u}$ in $L^1(G)$ and $|\mathit{\epsilon }({\mathbf{u}}_k)|(G)\to |\mathit{\epsilon }(\mathbf{u})|(G)$ for $k\to \infty$. Then we have

$\gamma ({\mathbf{u}}_k)\to \gamma (\mathbf{u})\mathit{\text{in}}{L}^1(\partial G,{\mathcal{H}}^{d-1}).$

c) If $\gamma (\mathbf{u})=0$ then we have

${\Vert \mathbf{u}\Vert }_{L^{d/(d-1)}(G)}\le c(d,G)|\mathit{\epsilon }(\mathbf{u})|(G).$

Let $G\subset {\mathbb{R}}^d$ be a bounded star-shaped domain with Lipschitz boundary ∂G. Then, for every $\mathbf{u}\in BD(G)$ there exists a sequence $({\mathbf{u}}_k)\subset \mathrm{BD}(G)\cap C_0^{\infty }(G)$ such that

${\mathbf{u}}_k\to \mathbf{u}\mathit{\text{in}}{L}^{d/(d-1)}(G),k\to \infty ,$

${\int }_G|\mathit{\epsilon }({\mathbf{u}}_k)|\mathrm{d}x\to |\mathit{\epsilon }(\mathbf{u})|(G)+{\int }_{\partial G}|\gamma (\mathbf{u})\odot \mathcal{N}|\mathrm{d}{\mathcal{H}}^{d-1},k\to \infty .$

Let $\mathbf{u}\in E_0^h(G)$. Then the field $\mathbf{w}:=\ln (1+|\mathbf{u}|)\mathbf{u}$ belongs to the space $\mathrm{BD}(G)$, and the total variation $|\mathit{\epsilon }(\mathbf{w})|(G)$ of w is bounded in terms of ${\Vert \mathit{\epsilon }(\mathbf{u})\Vert }_{L^h(G)}$, i.e. we have

$|\mathit{\epsilon }(\mathbf{w})|(G)\le C\left({\Vert \mathbf{u}\Vert }_{E_0^h(G)}\right).$

Consider first the case $\mathbf{u}\in C_0^{\infty }(G)$. Then it holds

$\mathit{\epsilon }(\mathbf{w})=\ln (1+|\mathbf{u}|)\mathit{\epsilon }(\mathbf{u})+\frac{1}{2}{({\mathbf{u}}^i{\partial }_j\ln (1+|\mathbf{u}|)+{\mathbf{u}}^j{\partial }_i\ln (1+|\mathbf{u}|))}_{1\le i,j\le n},$

hence

$|\mathit{\epsilon }(\mathbf{w})|\le \ln (1+|\mathbf{u}|)|\mathit{\epsilon }(\mathbf{u})|+c(n)\frac{|\mathbf{u}|}{1+|\mathbf{u}|}|\mathrm{\nabla }\mathbf{u}|.$

From Young's inequality for N-functions we get for s, $t\ge 0$

$h^{\prime }(t)s\le \tilde{h}(h^{\prime }(t))+h(s),$

$\tilde{h}$ denoting the conjugate function of h. Moreover we have

$\tilde{h}(h^{\prime }(t))=t{h}^{\prime }(t)-h(t)\le h(t).$

These inequalities imply

$\ln (1+|\mathbf{u}|)|\mathit{\epsilon }(\mathbf{u})|\le h^{\prime }(|\mathbf{u}|)|\mathit{\epsilon }(\mathbf{u})|\le h(|\mathbf{u}|)+h(|\mathit{\epsilon }(\mathbf{u})|,$

hence

${\int }_G|\mathit{\epsilon }(\mathbf{w})|\mathrm{d}x\le {\int }_{\mathrm{\Omega }}h(|\mathbf{u}|)\mathrm{d}x+{\int }_{G}h(|\mathit{\epsilon }(\mathbf{u})|)\mathrm{d}x+c(n){\int }_G|\mathrm{\nabla }\mathbf{u}|\mathrm{d}x.$

The quantity ${\int }_{G}h(|\mathit{\epsilon }(\mathbf{u})|)\mathrm{d}x$ can be estimated in terms of ${\Vert \mathit{\epsilon }(\mathbf{u})\Vert }_{L^h(G)}$ (and vice versa), to ${\int }_G|\mathrm{\nabla }\mathbf{u}|\mathrm{d}x$ we apply Lemma A.2.1, and finally observe that ${\int }_{G}h(|\mathbf{u}|)\mathrm{d}x$ is bounded e.g. by ${\int }_G|\mathbf{u}{|}^{d/d-1}\mathrm{d}x$ and this integral can be handled via (A.2.5). Altogether we have (A.2.10) for the smooth case.

If $\mathbf{u}\in E_0^h(G)$ is arbitrary, then we choose ${\mathbf{u}}_{\nu }\in C_0^{\infty }(G)$ such that ${\Vert \mathbf{u}-{\mathbf{u}}_{\nu }\Vert }_{E_0^h(G)}\to 0$ as $\nu \to \infty$. This in particular gives ${\Vert {\mathbf{u}}_{\nu }\Vert }_{E_0^h}\to {\Vert \mathbf{u}\Vert }_{E_0^h(G)}$, and (A.2.10) shows that

$\underset{\nu }{\sup }{\int }_G|\mathit{\epsilon }({\mathbf{w}}_{\nu })|\mathrm{d}x<\infty ,{\mathbf{w}}_{\nu }:=\ln (1+|{\mathbf{u}}_{\nu }|){\mathbf{u}}_{\nu }.$

If we apply (A.2.5) to ${\mathbf{u}}_{\nu }-\mathbf{u}$, we get ${\mathbf{u}}_{\nu }\to \mathbf{u}$ in $L^{d/(d-1)}(G)$, and for a suitable subsequence it holds ${\mathbf{u}}_{\nu }\to \mathbf{u}$ a.e., and therefore ${\mathbf{w}}_{\nu }\to \mathbf{w}$ a.e. By (A.2.11) and (A.2.5) we see that $\{{\mathbf{w}}_{\nu }\}$ is bounded sequence in $\mathrm{BD}(G)$, thus there is a strongly convergent subsequence in $L^1(G)$ (see Lemma A.2.1) which means that there exists $\tilde{\mathbf{w}}\in \mathrm{BD}(G)$ such that ${\mathbf{w}}_{\nu }\to \tilde{\mathbf{w}}$ in $L^1(G)$. The finiteness of $|\mathit{\epsilon }(\tilde{\mathbf{w}})|(G)$ follows by lower semi-continuity, i.e.

$|\mathit{\epsilon }(\tilde{\mathbf{w}})|(G)\le \underset{\nu \to \infty }{\lim \inf }{\int }_G|\mathit{\epsilon }({\mathbf{w}}_{\nu })|\mathrm{d}x.$

Clearly we have $\tilde{\mathbf{w}}=\mathbf{w}$, and (A.2.10) for w follows from (A.2.12) and the version of (A.2.10) for ${\mathbf{w}}_{\nu }$.  □

The embedding $E_0^h(G)\hookrightarrow L^{d/(d-1)}(G)$ is compact. More precisely, if ${\mathbf{u}}_{\nu }$ denotes a bounded sequence in $E_0^h(G)$, then there exists a subsequence ${\mathbf{u}}_{\nu }$ (not relabelled) and a function $\mathbf{u}\in E_0^h(G)$ such that ${\mathbf{u}}_{\nu }\to \mathbf{u}$ in $L^{d/(d-1)}(\mathrm{\Omega })$ and $\mathit{\epsilon }({\mathbf{u}}_{\nu })\rightharpoonup \mathit{\epsilon }(\mathbf{u})$ in $L^1(G)$ for $\nu \to \infty$.

Suppose that ${\sup }_{\nu \in \mathbb{N}}{\Vert {\mathbf{u}}_{\nu }\Vert }_{E_0^h(G)}<\infty$. From Lemma A.2.4 we deduce the existence of a field $\mathbf{u}\in L^1(G)$ such that

${\mathbf{u}}_{\nu }\to \mathbf{u}\text{in }{L}^1(G)\text{and a.e.},$

where here and in what follows we will pass to subsequences whenever this is necessary. According to the De La Vallée Poussin criterion for weak compactness in $L^1$ or by a theorem of Dunford and Pettis (cf. [9], Theorem 1.38) we get from

$\underset{\nu \in \mathbb{N}}{\sup }{\int }_G|\mathit{\epsilon }({\mathbf{u}}_{\nu })|\ln (1+|\mathit{\epsilon }({\mathbf{u}}_{\nu })|)\mathrm{d}x<\infty$

that $\mathit{\epsilon }({\mathbf{u}}_{\nu })\rightharpoonup :\mathit{\sigma }$ in $L^1(G)$, and clearly $\mathit{\sigma }=\mathit{\epsilon }(\mathbf{u})$. Moreover, by lower semi-continuity it holds

${\int }_{\mathrm{\Omega }}h(|\mathit{\epsilon }(\mathbf{u})|)\mathrm{d}x\le \underset{\nu \to \infty }{\lim \inf }{\int }_{G}h(|\mathit{\epsilon }({\mathbf{u}}_{\nu })|),$

so that u is an element of the space $E^{1,h}(G)$. In order to show $\mathbf{u}\in E_0^h(G)$, we follow the arguments of Frehse and Seregin [81]: since $\mathit{\epsilon }({\mathbf{u}}_{\nu })\rightharpoonup \mathit{\epsilon }(\mathbf{u})$ in $L^1(G)$ we can find a sequence $\{{\mathit{\sigma }}_{\mu }\}$, ${\mathit{\sigma }}_{\mu }$ being an element of the convex hull of $\{\mathit{\epsilon }({\mathbf{u}}_{\nu }):\nu \ge \mu \}$, such that ${\mathit{\sigma }}_{\mu }\to \mathit{\epsilon }(\mathbf{u})$ in $L^1(G)$. This follows from the well-known Banach–Saks lemma. We have

${\mathit{\sigma }}_{\mu }=\sum _{\nu =\mu }^{N(\mu )}{\lambda }_{\nu }^{\mu }\mathit{\epsilon }({\mathbf{u}}_{\nu }),\sum _{\nu =\mu }^{N(\mu )}{\lambda }_{\nu }^{\mu }=1,0\le {\lambda }_{\nu }^{\mu }\le 1$

with suitable coefficients ${\lambda }_{\nu }^{\mu }$ and integers $N(\mu )\ge \mu$. Let

${\bar{\mathbf{u}}}_{\mu }:=\sum _{\nu =\mu }^{N(\mu )}{\lambda }_{\nu }^{\mu }{\mathbf{u}}_{\nu }.$

These functions belong to $E_0^h(G)$ and satisfy

${\Vert {\bar{\mathbf{u}}}_{\mu }-\mathbf{u}\Vert }_{L^1(G)}\le \sum _{\nu =\mu }^{N(\mu )}{\lambda }_{\nu }^{\mu }{\Vert {\mathbf{u}}_{\nu }-\mathbf{u}\Vert }_{L^1(G)}\to 0,\mu \to \infty ,$

which is a consequence of (A.2.13). Moreover it holds

${\int }_G|\mathit{\epsilon }({\bar{\mathbf{u}}}_{\mu })|\mathrm{d}x={\int }_G|{\mathit{\sigma }}_{\mu }|\mathrm{d}x\to {\int }_G|\mathit{\epsilon }(\mathbf{u})|\mathrm{d}x,\mu \to \infty ,$

and according to Lemma A.2.2 b) these two convergences imply the $L^1$-convergence of the traces of ${\mathbf{u}}_{\mu }$ towards the trace of u. In conclusion $\mathbf{u}{|}_{\partial G}=0$, hence $\mathbf{u}\in E_0^h(G)$, and it remains to show that

${\mathbf{u}}_{\nu }\to \mathbf{u}\text{in }{L}^{d/(d-1)}(G)$

holds. From our assumption combined with (A.2.10) we get

$\underset{\nu \in \mathbb{N}}{\sup }{\int }_{\mathrm{\Omega }}|\mathit{\epsilon }({\mathbf{w}}_{\nu })|<\infty ,$

${\mathbf{w}}_{\nu }:=\ln (1+|{\mathbf{u}}_{\nu }|){\mathbf{u}}_{\nu }$, and (A.2.15) together with the first part of Lemma A.2.4 gives

$\underset{\nu \in \mathbb{N}}{\sup }{\Vert {\mathbf{w}}_{\nu }\Vert }_{L^{d/(d-1)}(G)}<\infty .$

Let $\mathrm{\Gamma }(t):=h{\left(t^{\frac{d-1}{d}}\right)}^{d/(d-1)},t\ge 0$. Then

$\frac{\mathrm{\Gamma }(t)}{t}={\left[\frac{h\left(t^{\frac{d-1}{d}}\right)}{t^{\frac{d-1}{d}}}\right]}^{\frac{d}{d-1}}⟶\infty ,t\to \infty ,$

and (compare (A.2.16))

${\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(|{\mathbf{u}}_{\nu }{|}^{\frac{d}{d-1}})\mathrm{d}x={\int }_{G}h{(|{\mathbf{u}}_{\nu }|)}^{\frac{d}{d-1}}\mathrm{d}x={\int }_{\mathrm{\Omega }}|{\mathbf{w}}_{\nu }{|}^{\frac{d}{d-1}}\mathrm{d}x\le c<\infty ,$

therefore $|{\mathbf{u}}_{\nu }{|}^{d/(d-1)}\rightharpoonup :g$ weakly in $L^1(G)$ by quoting the De La Vallée Poussin criterion one more time. By (A.2.13) we must have $g=|\mathbf{u}{|}^{d/(d-1)}$, since $|{\mathbf{u}}_{\nu }{|}^{d/(d-1)}\to |\mathbf{u}{|}^{d/(d-1)}$ a.e. on Ω. This in particular implies

${\Vert {\mathbf{u}}_{\nu }\Vert }_{L^{d/(d-1)}(G)}\to {\Vert \mathbf{u}\Vert }_{L^{d/(d-1)}(G)},\nu \to \infty ,$

where we combined (A.2.17) and (A.2.18) with Vitali's Theorem. At the same time it follows from

$\underset{\nu \in \mathbb{N}}{\sup }{\Vert {\mathbf{u}}_{\nu }\Vert }_{L^{d/(d-1)}(G)}<\infty$

and (A.2.13), that ${\mathbf{u}}_{\nu }\rightharpoonup \mathbf{u}$ in $L^{d/(d-1)}(G)$. Putting both convergences together, the Radon–Riesz lemma (cf. [89], p. 47, Proposition 3) gives our claim (A.2.14), and Theorem A.2.47 is proved.  □

The statement of Theorem A.2.47 remains valid, if the space $E_0^h(G)$ is replaced by the subclass $E_{0,\mathrm{div}}^h(G)$.