# 数学代写|泛函分析作业代写Functional Analysis代考|MA4551

#### Doug I. Jones

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## 数学代写|泛函分析作业代写Functional Analysis代考|Examples of Variational Formulations

We shall study now various variational formulations for a model diffusion-convection-reaction problem. We will use both versions of the Closed Range Theorem (for continuous and for closed operators) to demonstrate that different formulations are simultaneously well posed.

Diffusion-Convection-Reaction Problem. Given a domain $\Omega \subset \mathbb{R}^N, N \geq 1$, we wish to determine $u(x), x \in \bar{\Omega}$, that satisfies the boundary-value problem:
\left{\begin{aligned} -\left(a_{i j} u_{, j}\right){, i}+\left(b_i u\right){, i}+c u & =f & & \text { in } \Omega \ u & =0 & & \text { on } \Gamma_1 \ a_{i j} u_{, j} n_j-b_i n_i u & =0 & & \text { on } \Gamma_2 \end{aligned}\right.
Coefficients $a_{i j}(x)=a_{j i}(x), b_i(x), c(x)$ represent (anisotropic) diffusion, advection, and reaction, and $f$ stands for a source term. We are using the Einstein summation convention, the simplified, engineering notation for derivatives,
$$u_{, i} \stackrel{,}{=} \frac{\partial u}{\partial x_i}$$
and $n_i$ denote components of the unit outward vector on $\Gamma$. For instance, we can think of $u(x)$ as the temperature at point $x$ and $f(x)$ as representing a heat source (sink) at $x . \Gamma_1, \Gamma_2$ represent two disjoint parts of the boundary. For simplicity of the exposition, we will deal with homogeneous boundary conditions only.

## 数学代写|泛函分析作业代写Functional Analysis代考|Additional Facts about Sobolev Spaces

Additional Facts about Sobolev Spaces. We will need some additional fundamental facts about two energy spaces. The first is the already discussed classical $H^1$ Sobolev space consisting of all $L^2$-functions whose distributional derivatives are also functions, and they are $L^2$-integrable as well,
$$H^1(\Omega):=\left{u \in L^2(\Omega): \frac{\partial u}{\partial x_i} \in L^2(\Omega), i=1, \ldots, N\right}$$
The space is equipped with the norm,
$$|u|_{H^1}^2:=|u|^2+\sum_{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$
where $|\cdot|$ denotes the $L^2$-norm. The second term constitutes a seminorm on $H^1(\Omega)$ and will be denoted by
$$|u|{H^1}^2:=\sum{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$
The second space, $H(\operatorname{div}, \Omega)$, consists of all vector-valued $L^2$-integrable functions whose distributional divergence is also a function, and it is $L^2$-integrable,
$$H(\operatorname{div}, \Omega):=\left{\sigma=\left(\sigma_i\right)_{i=1}^N \in\left(L^2(\Omega)\right)^N: \operatorname{div} \sigma \in L^2(\Omega)\right}$$

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Examples of Variational Formulations

\left{\begin{aligned} -\left(a_{i j} u_{, j}\right){, i}+\left(b_i u\right){, i}+c u & =f & & \text { in } \Omega \ u & =0 & & \text { on } \Gamma_1 \ a_{i j} u_{, j} n_j-b_i n_i u & =0 & & \text { on } \Gamma_2 \end{aligned}\right.

$$u_{, i} \stackrel{,}{=} \frac{\partial u}{\partial x_i}$$
$n_i$表示$\Gamma$上单位向外向量的分量。例如，我们可以认为$u(x)$是$x$点的温度，$f(x)$代表热源(汇)，$x . \Gamma_1, \Gamma_2$代表边界的两个不相交的部分。为了说明的简单性，我们只处理齐次边界条件。

## 数学代写|泛函分析作业代写Functional Analysis代考|Additional Facts about Sobolev Spaces

$$H^1(\Omega):=\left{u \in L^2(\Omega): \frac{\partial u}{\partial x_i} \in L^2(\Omega), i=1, \ldots, N\right}$$

$$|u|{H^1}^2:=|u|^2+\sum{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$

$$|u|{H^1}^2:=\sum{i=1}^N\left|\frac{\partial u}{\partial x_i}\right|^2$$

$$H(\operatorname{div}, \Omega):=\left{\sigma=\left(\sigma_i\right)_{i=1}^N \in\left(L^2(\Omega)\right)^N: \operatorname{div} \sigma \in L^2(\Omega)\right}$$

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