# 物理代写|电磁学代写electromagnetism代考|PHYC20014

## 物理代写|电磁学代写electromagnetism代考|Helmholtz-Like Problem

Let $H$ and $V$ be two Hilbert spaces, such that $V$ is a vector subspace of $H$ with continuous imbedding $i_{V \rightarrow H}$. In what follows, we choose $H$ as the pivot space. Let $a(\cdot, \cdot)$ be a sesquilinear continuous form on $V \times V, A$ the corresponding operator defined at (4.4) with $V=W$, and $\lambda \in \mathbb{C} \backslash{0}$. Given $f \in V^{\prime}$, the Helmholtz-like problem to be solved is
$$\left{\begin{array}{l} \text { Find } u \in V \text { such that } \ \forall v \in V, a(u, v)+\lambda(u, v)H=\langle f, v\rangle . \end{array}\right.$$ Such problems are usually solved with the help of the Fredholm alternative. Theorem 4.5.1 (Helmholtz-Like Problem) Assume that the sesquilinear form a is such that $A$ is an isomorphism from $V$ to $V^{\prime}$, and that the canonical imbedding $i{V \rightarrow H}$ is compact. Then:

• either, for all $f \in V^{\prime}$, Problem (4.36) has one, and only one, solution $u$, which depends continuously on $f$;
• or, Problem (4.36) has solutions if, and only if, $f$ satisfies a finite number $n_\lambda$ of orthogonality conditions. Then, the space of solutions is affine, and the dimension of the corresponding linear vector space (i.e., the kernel) is equal to $n_\lambda$. Moreover, the part of the solution that is orthogonal to the kernel depends continuously on the data.

Proof Since the operator $A^{-1}$ is well-defined, one can replace the right-hand side with $a\left(A^{-1} f, v\right)$ in (4.36). Also, one can replace the second term as follows. We mention the imbedding $i_{V \rightarrow H}$ explicitly here, to write
$$\forall v \in V,(u, v)H=\left(i{V \rightarrow H} u, v\right)H=\left\langle i{V \rightarrow H} u, v\right\rangle=a\left(A^{-1} \circ i_{V \rightarrow H} u, v\right) .$$
So, Problem (4.36) equivalently rewrites
$$\left{\begin{array}{l} \text { Find } u \in V \text { such that } \ \left(I_V+\lambda A^{-1} \circ i_{V \rightarrow H}\right) u=A^{-1} f \text { in } V \end{array}\right.$$

## 物理代写|电磁学代写electromagnetism代考|Energy Conservation and Uniqueness

Let us consider that $\Omega=\mathbb{R}^3$ is made of a perfect medium (cf. Eqs. (1.18-1.21)), plus initial conditions at time $t=0$ (cf. (1.31)), i.e., $I=] 0,+\infty[$ :
$$\begin{gathered} \mathbb{C} \frac{\partial \boldsymbol{E}}{\partial t}-\operatorname{curl} \boldsymbol{H}=-\boldsymbol{J}, \quad t>0 \ \mu \frac{\partial \boldsymbol{H}}{\partial t}+\operatorname{curl} \boldsymbol{E}=0, \quad t>0 \end{gathered}$$

We also consider that $\Omega$ is an unbounded open subset of $\mathbb{R}^3$ of category (C2) equal to $\Omega=\mathbb{R}^3 \backslash \bar{O}$, where $O$ can be a perfectly conducting obstacle, as for the exterior problem, or the perfectly conducting device of interest, as for the interior problem (cf. Sect. 1.6.1). Or, we let $\Omega \subset \mathbb{R}^3$ be a domain made of a perfect medium, encased in a perfect conductor. We call this setting the cavity problem. In this case, we add boundary conditions on $\Gamma=\partial \Omega$ to (5.3)-(5.7):
\begin{aligned} \mu \boldsymbol{H} \cdot \boldsymbol{n}=0, & t>0 \ \boldsymbol{E} \times \boldsymbol{n}=0 . & t>0 \end{aligned}
Using the regularity results $(5.2)$ in space and time of the electromagnetic fields ${ }^1$ (and of the data $\boldsymbol{J}$ ), let us recover the energy conservation relation, starting from Ampère’s and Faraday’s laws. Above, $\S \in{\mathbb{C}, \mu}$ satisfies the following assumption: $\left{\begin{array}{l}\xi \text { is a real-valued, symmetric, measurable tensor field on } \Omega, \ \exists \xi_{-}, \xi_{+}>0, \forall \boldsymbol{X} \in \mathbb{C}^3, \xi_{-}|\boldsymbol{X}|^2 \leq 8 \boldsymbol{X} \cdot \overline{\boldsymbol{X}} \leq \xi_{+}|\boldsymbol{X}|^2 \text { a.e. in } \Omega .\end{array}\right.$

Remark 5.1.3 Obviously, one infers similar estimates involving the inverses of $\varepsilon_{-}, \varepsilon_{+}$(respectively of $\mu_{-}, \mu_{+}$) for the tensor $\mathbb{C}^{-1}$ (respectively $\mu^{-1}$ ). These assumptions will be frequently used throughout Chaps. $5,6,7$ and 8 . They include the case of an inhomogeneous medium $\left(\mathbb{c}=\varepsilon \mathbb{Z}_3, \mu=\mu \mathbb{I}_3\right)$.

## 物理代写|电磁学代写electromagnetism代考|Helmholtz-Like Problem

$\$ \$$\backslash left } Find u \in V such that \forall v \in V, a(u, v)+\lambda(u, v) H=\langle f, v\rangle. 【正确的。\\ 此类问题通常在 Fredholm 替代方案的帮助下得到解决。定理 4.5 .1 (类亥姆霍兹问题) 假设 倍半线性形式 a 是这样的 A 是来自的同构 V 至 V^{\prime} ，并且规范嵌入 i V \rightarrow H 紧凑。然后: • 或者，对于所有人 f \in V^{\prime} ，问题 (4.36) 有一个且只有一个解 u ，它持续依赖于 f; • 或者，问题 (4.36) 有解当且仅当， f 满足有限数 n_\lambda 的正交性条件。那么解的空间是仿射的，对应的线性 向量空间 (即核) 的维数等于 n_\lambda. 此外，解中与内核正交的部分持续依赖于数据。 证明自运营商 A^{-1} 是明确定义的，可以将右侧替换为 a\left(A^{-1} f, v\right) 在（4.36) 中。此外，可以如下萌换第二 项。我们提到嵌入 i_{V \rightarrow H} 在这里明确地写$$
\forall v \in V,(u, v) H=(i V \rightarrow H u, v) H=\langle i V \rightarrow H u, v\rangle=a\left(A^{-1} \circ i_{V \rightarrow H} u, v\right) .
$$因此，问题 (4.36) 等效地重写了 \ \$$
$\backslash$ left
Find $u \in V$ such that $\left(I_V+\lambda A^{-1} \circ i_{V \rightarrow H}\right) u=A^{-1} f$ in $V$
【正确的。
$\$ \$$## 物理代写|电磁学代写electromagnetism代考|Energy Conservation and Uniqueness 让我们考虑一下 \Omega=\mathbb{R}^3 由完美介质 (参见方程 (1.18-1.21) ) 加上初始条件组成 t=0 (参见 (1.31)) ，即， I=] 0,+\infty[:$$
\mathbb{C} \frac{\partial \boldsymbol{E}}{\partial t}-\operatorname{curl} \boldsymbol{H}=-\boldsymbol{J}, \quad t>0 \mu \frac{\partial \boldsymbol{H}}{\partial t}+\operatorname{curl} \boldsymbol{E}=0, \quad t>0
$$我们也认为 \Omega 是一个无界开子集 \mathbb{R}^3 类别 (C2) 等于 \Omega=\mathbb{R}^3 \backslash \bar{O} ，在哪里 O 对于外部问题，可以是一个完全 导电的障碍物，或者对于内部问题，可以是一个完全导电的设备（参见第 1.6 .1 节) 。或者，我们让 \Omega \subset \mathbb{R}^3 是一个由完美介质构成的领域，包䡕在完美导体中。我们称这种设置为空腔问题。在这种情况 下，我们在 \Gamma=\partial \Omega 至 (5.3) – (5.7) :$$
\mu \boldsymbol{H} \cdot \boldsymbol{n}=0, t>0 \boldsymbol{E} \times \boldsymbol{n}=0 . \quad t>0


myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: