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# 数学代写|泛函分析代写Functional Analysis代考|MATH510

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## 数学代写|泛函分析代写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,
$$u_{, i} \stackrel{\prime}{=} \frac{\partial u}{\partial x_i}$$
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.

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

The space is equipped with the norm,
$$|\sigma|_{H(\text { div })}^2:=|\sigma|^2+|\operatorname{div} \sigma|^2$$
where the $L^2$-norm of vector-valued functions is computed componentwise,
$$|\sigma|^2:=\sum_{i=1}^N\left|\sigma_i\right|^2$$
For both energy spaces, there exist trace operators that generalize the classical boundary trace for scalarvalued functions and boundary normal trace for vector-valued functions,
$$\left.u \rightarrow u\right|{\Gamma}, \quad \sigma \rightarrow \sigma_n=\left.\sum{i=1}^N \sigma_i\right|_{\Gamma} n_i$$

## 数学代写|泛函分析代写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.

$$u_{, i} \stackrel{\prime}{=} \frac{\partial u}{\partial x_i}$$
$n_i$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。

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

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

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

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

$$|\sigma|{H(\text { div })}^2:=|\sigma|^2+|\operatorname{div} \sigma|^2$$ 其中向量值函数的$L^2$ -范数是按分量计算的， $$|\sigma|^2:=\sum{i=1}^N\left|\sigma_i\right|^2$$

$$\left.u \rightarrow u\right|{\Gamma}, \quad \sigma \rightarrow \sigma_n=\left.\sum{i=1}^N \sigma_i\right|_{\Gamma} n_i$$

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