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# 数学代写|偏微分方程代考Partial Differential Equations代写|MTH3023 Boundaries of open sets

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Boundaries of open sets

When we analyze solutions of a PDE in the interior of their domain of definition, we can often consider domains that are arbitrary open sets and analyze the solutions in a sufficiently small ball. In order to analyze the behavior of solutions at a boundary, however, we typically need to assume that the boundary has some sort of smoothness. In this section, we define the smoothness of the boundary of an open set. We also explain briefly how one defines analytically the normal vector-field and the surface area measure on a smooth boundary.

In general, the boundary of an open set may be complicated. For example, it can have nonzero Lebesgue measure.

ExAmPLE 1.32. Let $\left{q_i: i \in \mathbb{N}\right}$ be an enumeration of the rational numbers $q_i \in(0,1)$. For each $i \in \mathbb{N}$, choose an open interval $\left(a_i, b_i\right) \subset(0,1)$ that contains $q_i$, and let
$$\Omega=\bigcup_{i \in \mathbb{N}}\left(a_i, b_i\right) .$$
The Lebesgue measure of $|\Omega|>0$ is positive, but we can make it as small as we wish; for example, choosing $b_i-a_i=\epsilon 2^{-i}$, we get $|\Omega| \leq \epsilon$. One can check that $\partial \Omega=[0,1] \backslash \Omega$. Thus, if $|\Omega|<1$, then $\partial \Omega$ has nonzero Lebesgue measure.

Moreover, an open set, or domain, need not lie on one side of its boundary (we say that $\Omega$ lies on one side of its boundary if $\bar{\Omega} \circ=\Omega$ ), and corners, cusps, or other singularities in the boundary cause analytical difficulties.
EXAMPLE 1.33. The unit disc in $\mathbb{R}^2$ with the nonnegative $x$-axis removed,
$$\Omega=\left{(x, y) \in \mathbb{R}^2: x^2+y^2<1\right} \backslash\left{(x, 0) \in \mathbb{R}^2: 0 \leq x<1\right},$$
does not lie on one side of its boundary.

## 数学代写|偏微分方程代考Partial Differential Equations代写|Open sets in the plane

Open sets in the plane. A simple closed curve, or Jordan curve, $\Gamma$ is a set in the plane that is homeomorphic to a circle. That is, $\Gamma=\gamma(\mathbb{T})$ is the image of a one-to-one continuous map $\gamma: \mathbb{T} \rightarrow \mathbb{R}^2$ with continuous inverse $\gamma^{-1}: \Gamma \rightarrow \mathbb{T}$. (The requirement that the inverse is continuous follows from the other assumptions.) According to the Jordan curve theorem, a Jordan curve divides the plane into two disjoint connected open sets, so that $\mathbb{R}^2 \backslash \Gamma=\Omega_1 \cup \Omega_2$. One of the sets (the ‘interior’) is bounded and simply connected. The interior region of a Jordan curve is called a Jordan domain.

Example 1.37. The slit $\operatorname{disc} \Omega$ in Example 1.33 is not a Jordan domain. For example, its boundary separates into two nonempty connected components when the point $(1,0)$ is removed, but the circle remains connected when any point is removed, so $\partial \Omega$ cannot be homeomorphic to the circle.

ExAmple 1.38. The interior $\Omega$ of the Koch, or ‘snowflake,’ curve is a Jordan domain. The Hausdorff dimension of its boundary is strictly greater than one. It is interesting to note that, despite the irregular nature of its boundary, this domain has the property that every function in $W^{k, p}(\Omega)$ with $k \in \mathbb{N}$ and $1 \leq p<\infty$ can be extended to a function in $W^{k, p}\left(\mathbb{R}^2\right)$.

If $\gamma: \mathbb{T} \rightarrow \mathbb{R}^2$ is one-to-one, $C^1$, and $|D \gamma| \neq 0$, then the image of $\gamma$ is the $C^1$ boundary of the open set which it encloses. The condition that $\gamma$ is one-toone is necessary to avoid self-intersections (for example, a figure-eight curve), and the condition that $|D \gamma| \neq 0$ is necessary in order to ensure that the image is a $C^1$-submanifold of $\mathbb{R}^2$.

EXAMPLE 1.39. The curve $\gamma: t \mapsto\left(t^2, t^3\right)$ is not $C^1$ at $t=0$ where $D \gamma(0)=0$.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Boundaries of open sets

$$\Omega=\bigcup_{i \in \mathbb{N}}\left(a_i, b_i\right) .$$

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## 数学代写|偏微分方程代米Partial Differential Equations代写|Open sets in the plane

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## MATLAB代写

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