0
$\begingroup$

I have a question about Sobolev space.

Let $\Omega$ be an open subset of $\mathbb{R}^{d}$,

we consider the Sobolev space

$H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1,\ldots,n \right\}$

with norm

$\|u\|^{2}_{H^{1}(\Omega)}:=\|u\|^{2}_{L^{2}(\Omega)}+\sum_{j=1}^{n}\|D_{j}u\|_{L^{2}(\Omega)}^{2}$,

where $D_{j}u=\partial u/ \partial x_{j}$ is the distributional deriavtive. Moreover, we let

$X:=\left\{ u|{}_{\Omega} \in H^{1}(\Omega): u \in C_{c}(\bar{\Omega})\right\}, $ $Y:=\left\{ u|{}_{\Omega} \in H^{1}(\Omega): u \in C(\bar{\Omega})\right\},$

where $C_{c}(\bar{\Omega})$ denotes the space of all continuous real valued functions on $\bar{\Omega}$ with compact support and $C(\bar{\Omega})$ denotes the space of all continuous real valued functions on $\bar{\Omega}$.

My question

For any $\epsilon>0$, $f \in Y$, there exists $\varphi \in X$ s.t. $\|f-\varphi\|_{H^{1}(\Omega)}<\epsilon$ ?

My attempt 1

By Tietze extension theorem, we can find $F \in C(\mathbb{R}^{d})$ such that $f=F$ on $\bar{\Omega}$. Let $J_{\delta}F=\int_{\mathbb{R}^{d}}j_{\delta}(x-y)F(y)\,dy$, where $j_{\delta}$ is mollifier. Then $F_{\delta} \in C_{c}^{\infty}(\mathbb{R}^{d})$ and $J_{\delta}F \to f$ in $L^{2}(\Omega)$ as $\delta \to 0$. Futhermore, since $f \in H^{1}(\Omega)$, \begin{align*} \frac{\partial}{\partial x_{i}}J_{\delta}F-\frac{\partial f}{\partial x_{i}}&=J_{\delta}\frac{\partial F}{\partial x_{i}}-\frac{\partial F}{\partial x_{i}}\quad \text{on } \Omega \\ & \to 0 \quad \text{in } L^{2}(\Omega). \end{align*} Therefore $\|J_{\delta}F -f\|_{H^{1}(\Omega)} \to 0$ as $\delta \to 0$. But I don't know $J_{\delta}F \left| \right. _{\Omega} \in C_{c}(\bar{\Omega})$. Can we show $J_{\delta}F \left| \right. _{\Omega} \in C_{c}(\bar{\Omega})$ ?

My attempt 2

If there exist open subset $O$ and compact subset $K$ of $\mathbb{R}^{d}$ s.t. $O \subset K \subset \Omega$ and $\|f\|_{H^{1}(\Omega \setminus K)}<\epsilon$, we take $\psi \in C_{c}^{\infty}(\mathbb{R}^{d})$ s.t. $\psi=1$ on $O$ and $\psi=0$ on $\mathbb{R}^{d} \setminus K$.

Then \begin{align*} \|f \psi -f\|_{H^{1}(\Omega)} &\leq \|f \psi -f\|_{H^{1}(O)}+\|f \psi -f\|_{H^{1}(K \setminus O)}+\|f \psi -f\|_{H^{1}(\Omega \setminus K)} \\ &=\|f -f\|_{H^{1}(O)}+\|f \psi -f\|_{H^{1}(K \setminus O)}+\|0 -f\|_{H^{1}(\Omega \setminus K)} \\ &\leq 0+\|f \psi -f\|_{H^{1}(K \setminus O)}+\epsilon. \end{align*}

But I don't know how to deal with $\|f \psi -f\|_{H^{1}(K \setminus O)}$ and existence of such $O,K$.

If you know, please tell me. Thank you in advance.

$\endgroup$
  • $\begingroup$ For $u$ that is zero on the boundary, the result follows easily. I guess trace theory deals with this. $\endgroup$ – Mambo Apr 29 '16 at 10:27
  • $\begingroup$ Thank you for your reply. But I am not familiar with trace theory. Please tell me the details about that. $\endgroup$ – sharpe Apr 29 '16 at 10:39
  • $\begingroup$ Which book are you reading this from? There is a book by Brezis on Sobolev spaces. $\endgroup$ – Mambo Apr 29 '16 at 13:11
  • $\begingroup$ Thank you for your reply. This is my original problem. $\endgroup$ – sharpe Apr 29 '16 at 13:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.