# 数学代写|偏微分方程代写partial difference equations代考|MATH2415

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## 数学代写|偏微分方程代写partial difference equations代考|Mollifiers

In this section we will introduce an important technique which will permit us to approximate integrable functions by smooth functions. It is based on convolution with smooth functions.

Let $\Omega$ be an open set in $\mathbb{R}^d$. We say that the function $u: \Omega \rightarrow \mathbb{R}$ has compact support if there exists a compact set $K \subset \Omega$ such that $u(x)=0$ for all $x \in \Omega \backslash K$. If $u$ is also continuous, then as before (cf. (2.1)) we call the set
$$\operatorname{supp} u:=\overline{{x \in \Omega: u(x) \neq 0}}$$
the support of $u$. We denote by $C_c(\Omega)$ the space of all continuous real-valued functions on $\Omega$ with compact support, cf. (3.105). For $k \in \mathbb{N}$ we also set
$$C_c^k(\Omega):=C^k(\Omega) \cap C_c(\Omega), \quad C_c^0(\Omega):=C_c(\Omega) .$$
The space
$$\mathcal{D}(\Omega):=C_c^{\infty}(\Omega):=C^{\infty}(\Omega) \cap C_c(\Omega)$$
plays a special role; its elements are called test functions on $\Omega$. The following special test function $\varrho \in \mathcal{D}\left(\mathbb{R}^d\right)$ will be of particular importance:
$$\varrho(x):= \begin{cases}c \exp \left(\frac{1}{|x|^2-1}\right), & \text { if }|x|<1, \ 0, & \text { if }|x| \geq 1,\end{cases}$$ where $c>0$ is chosen in such a way that
$$\int_{\mathbb{R}^d} \varrho(x) d x-1 .$$

## 数学代写|偏微分方程代写partial difference equations代考|Sobolev spaces on Ω ⊆

After the preparatory work undertaken in Section $6.1$ we can now introduce Sobolev spaces on general domains in $\mathbb{R}^d$. So let $\Omega$ be an open set in $\mathbb{R}^d$; then as in Chapter 5 we shall define weak derivatives of functions on $\Omega$ via integration by parts. To start with, we will suppose that $f$ is a function which is continuously differentiable in the classical sense, that is, $f \in C^1(\Omega)$ with partial derivatives $\frac{\partial f}{\partial x_j}, j=1, \ldots, d$.
Lemma 6.12 (Integration by parts) For $f \in C^1(\Omega)$ and $\varphi \in C_c^1(\Omega)$ we have
$$-\int_{\Omega} f(x) \frac{\partial \varphi}{\partial x_j}(x) d x=\int_{\Omega} \frac{\partial f}{\partial x_j} \varphi(x) d x$$
Proof lst case: $\Omega=\mathbb{R}^d$. In this case, we have
\begin{aligned} -\int_{\mathbb{R}^d} f(x) \frac{\partial \varphi}{\partial x_j}(x) d x & =-\int_{\mathbb{R}} \cdots \int_{\mathbb{R}} f\left(x_1, \ldots, x_d\right) \frac{\partial \varphi}{\partial x_j}\left(x_1, \ldots, x_d\right) d x_1 \cdots d x_d \ & =\int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \frac{\partial f}{\partial x_j}\left(x_1, \ldots, x_d\right) \varphi\left(x_1, \ldots, x_d\right) d x_1 \cdots d x_d \end{aligned}
by integration by parts in the $j$ th integral.
2nd case: Now suppose $\Omega$ is arbitrary. Choose $U \Subset \Omega$ such that $\operatorname{supp} \varphi \subset U$, and also choose $\eta \in \mathcal{D}(\Omega)$ such that $\eta=1$ on $\operatorname{supp} \varphi$ and $\operatorname{supp} \eta \subset U$ (see Lemma 6.7). Then $\widetilde{f \eta} \in C^1\left(\mathbb{R}^d\right.$ ) (where, as in the previous section, $\widetilde{f \eta}=(f \eta)^{\sim}$ denotes the extension of the function $f \eta$ by zero in accordance with (6.20)). By the 1st case, with the relevant extensions,
\begin{aligned} -\int_{\Omega} f(x) \frac{\partial \varphi}{\partial x_j}(x) d x & =-\int_{\mathbb{R}^d} \widetilde{f \eta}(x) \frac{\partial \varphi}{\partial x_j}(x) d x \ & =\int_{\mathbb{R}^d} \frac{\partial(\widetilde{f \eta})(x)}{\partial x_j} \varphi(x) d x=\int_{\Omega} \frac{\partial f}{\partial x_j}(x) \varphi(x) d x, \end{aligned}
since $\eta=1$ on $\operatorname{supp} \varphi$ and $\operatorname{supp} \varphi \subset U \Subset \Omega$.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Mollifiers

$$\operatorname{supp} u:=\overline{x \in \Omega: u(x) \neq 0}$$

$$C_c^k(\Omega):=C^k(\Omega) \cap C_c(\Omega), \quad C_c^0(\Omega):=C_c(\Omega) .$$

$$\mathcal{D}(\Omega):=C_c^{\infty}(\Omega):=C^{\infty}(\Omega) \cap C_c(\Omega)$$

$$\varrho(x):=\left{c \exp \left(\frac{1}{|x|^2-1}\right), \quad \text { if }|x|<1,0, \quad \text { if }|x| \geq 1\right.$$ 在哪里 $c>0$ 以这样的方式选择
$$\int_{\mathbb{R}^d} \varrho(x) d x-1$$

## 数学代写|偏微分方程代写partial difference equations代考|Sobolev spaces on Ω ⊆

$$-\int_{\Omega} f(x) \frac{\partial \varphi}{\partial x_j}(x) d x=\int_{\Omega} \frac{\partial f}{\partial x_j} \varphi(x) d x$$

$$-\int_{\mathbb{R}^d} f(x) \frac{\partial \varphi}{\partial x_j}(x) d x=-\int_{\mathbb{R}} \cdots \int_{\mathbb{R}} f\left(x_1, \ldots, x_d\right)$$

$$-\int_{\Omega} f(x) \frac{\partial \varphi}{\partial x_j}(x) d x=-\int_{\mathbb{R}^d} \widetilde{f \eta}(x) \frac{\partial \varphi}{\partial x_j}(x) d x$$

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