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# 数学代写|偏微分方程代考Partial Differential Equations代写|Variational formulation

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Variational formulation

Definition 4.2 of a weak solution in is closely connected with the variational formulation of the Dirichlet problem for Poisson’s equation. To explain this connection, we first summarize some definitions of the differentiability of functionals (scalar-valued functions) acting on a Banach space.

Definition 4.3. A functional $J: X \rightarrow \mathbb{R}$ on a Banach space $X$ is differentiable at $x \in X$ if there is a bounded linear functional $A: X \rightarrow \mathbb{R}$ such that
$$\lim _{h \rightarrow 0} \frac{|J(x+h)-J(x)-A h|}{|h|_X}=0 .$$
If $A$ exists, then it is unique, and it is called the derivative, or differential, of $J$ at $x$, denoted $D J(x)=A$.

This definition expresses the basic idea of a differentiable function as one which can be approximated locally by a linear map. If $J$ is differentiable at every point of $X$, then $D J: X \rightarrow X^$ maps $x \in X$ to the linear functional $D J(x) \in X^$ that approximates $J$ near $x$.

A weaker notion of differentiability (even for functions $J: \mathbb{R}^2 \rightarrow \mathbb{R}-$ see Example 4.4) is the existence of directional derivatives
$$\delta J(x ; h)=\lim {\epsilon \rightarrow 0}\left[\frac{J(x+\epsilon h)-J(x)}{\epsilon}\right]=\left.\frac{d}{d \epsilon} J(x+\epsilon h)\right|{\epsilon=0} .$$
If the directional derivative at $x$ exists for every $h \in X$ and is a bounded linear functional on $h$, then $\delta J(x ; h)=\delta J(x) h$ where $\delta J(x) \in X^*$. We call $\delta J(x)$ the Gâteaux derivative of $J$ at $x$. The derivative $D J$ is then called the Fréchet derivative to distinguish it from the directional or Gâteaux derivative. If $J$ is differentiable at $x$, then it is Gâteaux-differentiable at $x$ and $D J(x)=\delta J(x)$, but the converse is not true.
EXAMPLE 4.4. Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $f(0,0)=0$ and
$$f(x, y)=\left(\frac{x y^2}{x^2+y^4}\right)^2 \quad \text { if }(x, y) \neq(0,0)$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|The space $H^{-1}(\Omega)$

The negative order Sobolev space $H^{-1}(\Omega)$ can be described as a space of distributions on $\Omega$.

ThEOrEM 4.7. The space $H^{-1}(\Omega)$ consists of all distributions $f \in \mathcal{D}^{\prime}(\Omega)$ of the form
$$f=f_0+\sum_{i=1}^n \partial_i f_i \quad \text { where } f_0, f_i \in L^2(\Omega)$$
These distributions extend uniquely by continuity from $\mathcal{D}(\Omega)$ to bounded linear functionals on $H_0^1(\Omega)$. Moreover,
$$|f|_{H^{-1}(\Omega)}=\inf \left{\left(\sum_{i=0}^n \int_{\Omega} f_i^2 d x\right)^{1 / 2}: \text { such that } f_0, f_i \text { satisfy (4.8) }\right}$$
Proof. First suppose that $f \in H^{-1}(\Omega)$. By the Riesz representation theorem there is a function $g \in H_0^1(\Omega)$ such that
(4.10) $\langle f, \phi\rangle=(g, \phi){H_0^1} \quad$ for all $\phi \in H_0^1(\Omega)$. Here, $(\cdot, \cdot){H_0^1}$ denotes the standard inner product on $H_0^1(\Omega)$,
$$(u, v){H_0^1}=\int{\Omega}(u v+D u \cdot D v) d x$$
Identifying a function $g \in L^2(\Omega)$ with its corresponding regular distribution, restricting $f$ to $\phi \in \mathcal{D}(\Omega) \subset H_0^1(\Omega)$, and using the definition of the distributional derivative, we have
\begin{aligned} \langle f, \phi\rangle & =\int_{\Omega} g \phi d x+\sum_{i=1}^n \int_{\Omega} \partial_i g \partial_i \phi d x \ & =\langle g, \phi\rangle+\sum_{i=1}^n\left\langle\partial_i g, \partial_i \phi\right\rangle \ & =\left\langle g-\sum_{i=1}^n \partial_i g_i, \phi\right\rangle \quad \text { for all } \phi \in \mathcal{D}(\Omega), \end{aligned}
where $g_i=\partial_i g \in L^2(\Omega)$. Thus the restriction of every $f \in H^{-1}(\Omega)$ from $H_0^1(\Omega)$ to $\mathcal{D}(\Omega)$ is a distribution
$$f=g-\sum_{i=1}^n \partial_i g_i$$
of the form (4.8). Also note that taking $\phi=g$ in $(4.10)$, we get $\langle f, g\rangle=|g|_{H_0^1}^2$, which implies that
$$|f|_{H^{-1}} \geq|g|_{H_0^1}=\left(\int_{\Omega} g^2 d x+\sum_{i=1}^n \int_{\Omega} g_i^2 d x\right)^{1 / 2}$$
which proves inequality in one direction of (4.9).

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Variational formulation

4.3.定义Banach空间$X$上的一个泛函$J: X \rightarrow \mathbb{R}$在$x \in X$是可微的，如果存在一个有界线性泛函$A: X \rightarrow \mathbb{R}$，使得
$$\lim _{h \rightarrow 0} \frac{|J(x+h)-J(x)-A h|}{|h|_X}=0 .$$

$$\delta J(x ; h)=\lim {\epsilon \rightarrow 0}\left[\frac{J(x+\epsilon h)-J(x)}{\epsilon}\right]=\left.\frac{d}{d \epsilon} J(x+\epsilon h)\right|{\epsilon=0} .$$

$$f(x, y)=\left(\frac{x y^2}{x^2+y^4}\right)^2 \quad \text { if }(x, y) \neq(0,0)$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|The space $H^{-1}(\Omega)$

$$f=f_0+\sum_{i=1}^n \partial_i f_i \quad \text { where } f_0, f_i \in L^2(\Omega)$$

$$|f|{H^{-1}(\Omega)}=\inf \left{\left(\sum{i=0}^n \int_{\Omega} f_i^2 d x\right)^{1 / 2}: \text { such that } f_0, f_i \text { satisfy (4.8) }\right}$$

(4.10) $\langle f, \phi\rangle=(g, \phi){H_0^1} \quad$为所有$\phi \in H_0^1(\Omega)$。其中，$(\cdot, \cdot){H_0^1}$表示$H_0^1(\Omega)$上的标准内积;
$$(u, v){H_0^1}=\int{\Omega}(u v+D u \cdot D v) d x$$

\begin{aligned} \langle f, \phi\rangle & =\int_{\Omega} g \phi d x+\sum_{i=1}^n \int_{\Omega} \partial_i g \partial_i \phi d x \ & =\langle g, \phi\rangle+\sum_{i=1}^n\left\langle\partial_i g, \partial_i \phi\right\rangle \ & =\left\langle g-\sum_{i=1}^n \partial_i g_i, \phi\right\rangle \quad \text { for all } \phi \in \mathcal{D}(\Omega), \end{aligned}

$$f=g-\sum_{i=1}^n \partial_i g_i$$

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