Posted on Categories:Ordinary Differential Equations, 常微分方程, 数学代写

# 数学代写|常微分方程代考Ordinary Differential Equations代写|MA26600 Diferential Operators with Smooth Coefcients

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Diferential Operators with Smooth Coefcients

A linear partial differential operator with coefficients $c_\alpha \in C^{\infty}(\Omega$ ) (in the sequel, often abbreviated to differential operator or even to $P D O$ – we shall practically never come across a linear partial differential equation, i.e., a $P D E$, whose coefficients are not smooth) acts on functions $f \in C^{\infty}(\Omega)$ :
$$P\left(x, \partial_x\right) f(x)=\sum_\alpha c_\alpha(x) \partial_x^\alpha f(x) \text { or } P\left(x, \mathrm{D}x\right) f=\sum\alpha c_\alpha(x) \mathrm{D}x^\alpha f(x)$$ The sums in (1.3.1) are locally finite: in every compact subset of $\Omega$ only finitely many $c\alpha$ do not vanish identically; $f \mapsto P(x$, D) $f$ is a linear continuous endomorphism of $C^{\infty}(\Omega)$. It is not difficult to prove that the obvious inclusion
$$\operatorname{supp} P(x, \mathrm{D}) f \subset \operatorname{supp} f$$
characterizes linear PDOs among all continuous linear endomorphisms of $C^{\infty}(\Omega)$ [actually among all linear endomorphisms of $C^{\infty}(\Omega)$, but this is more difficult to prove; see [Peetre, 1960]]. Formula (1.3.2) implies that $P(x, \mathrm{D})$ maps $C_{\mathrm{c}}^{\infty}(\Omega)$ into itself.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Defnitions. Support and singular support

Let $\Omega$ be an open subset of $\mathbb{R}^n$, as before. If $u$ is a complex-valued linear functional on the vector space $C_{\mathrm{c}}^{\infty}(\Omega)$, i.e., if $u$ is a linear map $C_{\mathrm{c}}^{\infty}(\Omega) \longrightarrow \mathbb{C}$, we denote by $\langle u, \varphi\rangle$ its evaluation at the test-function $\varphi \in C_{\mathrm{c}}^{\infty}(\Omega)$. The linear functional $u$ is a distribution in $\Omega$ if $\left\langle u, \varphi_j\right\rangle \rightarrow 0$ whenever the sequence $\left{\varphi_j\right}_{j=0,1,2, \ldots} \subset \mathcal{C}_{\mathrm{c}}^{\infty}(\Omega)$ converges to zero in the following sense:
(•) all derivatives $\partial^\alpha \varphi_j$ converge uniformly to zero and there is a compact set $K \subset \Omega$ such that $\operatorname{supp} \varphi_j \subset K$ whatever $j$.

The space of distributions in $\Omega$ is denoted by $\mathcal{D}^{\prime}(\Omega)$. The restriction of a distribution $u \in \mathcal{D}^{\prime}(\Omega)$ to an open subset $\Omega^{\prime}$ of $\Omega$ is simply the restriction of the linear functional $u$ to the linear subspace $C_{\mathrm{c}}^{\infty}\left(\Omega^{\prime}\right)$ of $C_{\mathrm{c}}^{\infty}(\Omega)$. By using partitions of unity in $C_{\mathrm{c}}^{\infty}(\Omega)$ it is readily proved that there is a smallest closed subset of $\Omega$, called the support of $u$ and denoted by supp $u$, such that $u$ vanishes (“identically”) in $\Omega \backslash F$. The subspace of distributions in $\Omega$ that have compact support (contained in $\Omega$ ) is denoted by $\mathcal{E}^{\prime}(\Omega)$; it can be identified with the dual of $C^{\infty}(\Omega)$.

The convergence of a sequence of distributions $u_j\left(j \in \mathbb{Z}{+}\right)$is to be understood in the “weak sense”: $u_j \rightarrow 0$ if $\left\langle u_j, \varphi\right\rangle \rightarrow 0$ for each $\varphi \in C{\mathrm{c}}^{\infty}(\Omega)$. For $u_j \in \mathcal{E}^{\prime}(\Omega)$ to converge to zero in $\mathcal{E}^{\prime}(\Omega)$ it is moreover required that there be a compact set $K \subset \Omega$ such that $\operatorname{supp} u_j \subset K$ for all $j$.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Diferential

Operators with Smooth Coefcients 微分方程，即 $P D E$ ，其系数不平滑) 作用于函数 $f \in C^{\infty}(\Omega)$ :
$$P\left(x, \partial_x\right) f(x)=\sum_\alpha c_\alpha(x) \partial_x^\alpha f(x) \text { or } P(x, \mathrm{D} x) f=\sum \alpha c_\alpha(x) \mathrm{D} x^\alpha f(x)$$
(1.3.1) 中的和是局部陏限的: 在 $\Omega$ 只有有限多 $c \alpha$ 不要完洞失; $f \mapsto P(x, \mathrm{D}) f$ 是线侏连续目同态 $C^{\infty}(\Omega)$. 不难证明明显包含 $\operatorname{supp} P(x, \mathrm{D}) f \subset \operatorname{supp} f$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Defnitions.

(•) 所有导数 $\partial^\alpha \varphi_j$ 一致收敛于零且存在紧集 $K \subset \Omega$ 这样supp $\varphi_j \subset K$ 任何 $j$.

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## MATLAB代写

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