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# 数学代写|谱几何代写Spectral Geometry代考|MIT6.838 Elementary surgery

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## 数学代写|谱几何代写Spectral Geometry代考|Elementary surgery

We will explain in detail the following fact (in the context of the Dirichlet boundary conditions): A small hole in a domain does not affect the spectrum “too much”(see (14) below). Let $\Omega \subset \mathbb{R}^n$ a domain with smooth boundary. Let $x \in \Omega$, and $B(x, \epsilon)$ the ball of radius $\epsilon$ centered at the point $x$ ( $\epsilon$ is chosen small enough such that $B(x, \epsilon) \subset \Omega$ ). We denote by $\Omega_\epsilon$ the subset $\Omega-B(x, \epsilon)$, that is we make a hole of radius $\epsilon$ on $\Omega$.

Theorem 17. We consider the two domains $\Omega$ and $\Omega_\epsilon$ with the Dirichlet boundary conditions, and denote by $\left{\lambda_k\right}_{k=1}^{\infty}$ and by $\left{\lambda_k(\epsilon)\right}_{k=1}^{\infty}$ their respective spectrum.
Then, for each $k$, we have
$$\lim _{\epsilon \rightarrow 0} \lambda_k(\epsilon)=\lambda_k$$
Before showing this in detail, let us make a few remarks.
Remark 18. 1. The convergence is not uniform in $k$.

This result is a very specific part of a much more general facts. We get this type of results on manifolds, with subset much more general than balls, for example tubular neighborhood of submanifolds of codimension greater than 1. For the interested reader, we refer to the paper of Courtois [13] and to the book of Chavel [11].

We can get very precise asymptotic estimates of $\lambda_k(\epsilon)$ in terms of $\epsilon$. Again, we refer to [13] and references therein.

We have also the convergence of the eigenspace associated to $\lambda_k(\epsilon)$ to the eigenspace associated to $\lambda_k$, but this has to be defined precisely: a problem occurs if the multiplicity of $\lambda_k$ is not equal to the multiplicity of $\lambda_k(\epsilon)$, see [13].

As a consequence of the monotonicity, the same result is true if we excise a family of domains $V_\epsilon$ contained in a ball of radius $\epsilon \rightarrow 0$ : because $\Omega_\epsilon \subset \Omega-V_\epsilon$ we have $\lambda_k\left(\Omega-V_\epsilon\right) \leq \lambda_k\left(\Omega_\epsilon\right) \rightarrow \lambda_k(\Omega)$ and also $\lambda_k\left(\Omega-V_\epsilon\right) \geq \lambda_k(\Omega)$.

A similar result is true for a hole and Neumann boundary condition. The proof is more difficult and cannot be extended to the domains contained in a ball.

## 数学代写|谱几何代写Spectral Geometry代考|Nodal domains

On different pictures or video, it seems to appear that the eigenfunction(s) of the eigenvalue $\lambda_k$ on any domain $\Omega$ becomes more and more complicated as $k$ increase The simplest example is the interval $[0, L]$ where we have seen that the $\mathrm{k}$ th eigenvalue of the Dirichlet problem was $\frac{k^2 \pi^2}{L^2}$, and the eigenfunction corresponding to $\lambda_k$ was $f_k(x)=\sin \frac{k \pi}{L} x$.

This is the object of this part of the lecture to elaborate a little around this. I follow mainly the book of Chavel [11], p. 19-25.

Definition 19. Let $\Omega \subset \mathbb{R}^n$ a domain and $f: \Omega \rightarrow \mathbb{R}$ a continuous function. The nodal set of $f$ is the set $\left{f^{-1}(0)\right}$ and a nodal domain of $f$ is one connected component of $\bar{\Omega}-\left{f^{-1}(0)\right}$.

In the sequel, I will describe a situation for the Dirichlet eigenvalues. The situation for the Neumann problem is often similar, but not always. I can be more precise about what we see on the video around Chladni plates: the sound oblige the plate to vibrate and the vibration of the plate correspond to an eigenfunction. What we see is the place where the plate does not vibrate, and this correspond to the nodal set of the corresponding eigenfuctions. The impression is that for large frequence (large eigenvalue) the nodal line is more and more complicated and tends to be “everywhere” on the plate. moreover, we have the impression that the number of nodal domains tends to increase with $k$. This is what we want to clarify thanks to Theorem 20 and 21 .

## 数学代写|谱几何代写Spectral Geometry代考|Elementary surgery

〈left 缺少或无法识别的分隔符 他们各自的频谱。

$$\lim {\epsilon \rightarrow 0} \lambda_k(\epsilon)=\lambda_k$$ 在详细说明之前，让我们做一些评论。 备注 18. 1. 收敛在 $k$. 这个结果是电普遍的事实中非常具体的一部分。我们在流形上得到了这种类型的结果，子集比球更普遍，例如余维大于 1 的子流形 的管状邻域。对于感兴趣的读者，我们参考 Courtois [13] 的论文和 Chavel 的书[11]. 我们可以获得非常精确的渐近估计 $\lambda_k(\epsilon)$ 按照 $\epsilon$. 同样，我们参考了 [13] 和其中的参考文献。 我们也有相关的特征空间的收敛 $\lambda_k(\epsilon)$ 到关联的特征空间 $\lambda_k$ ，但这必须精确定义: 如果多重性 $\lambda_k$ 不等于的重数 $\lambda_k(\epsilon)$ ，参见 [13]。 作为单调性的结果，如果我们切除一个域族，同样的结果也是正确的 $V\epsilon$ 包含在一个半径的球中 $\epsilon \rightarrow 0$ : 因为 $\Omega_\epsilon \subset \Omega-V_\epsilon$ 我们有 $\lambda_k\left(\Omega-V_\epsilon\right) \leq \lambda_k\left(\Omega_\epsilon\right) \rightarrow \lambda_k(\Omega)$ 并且 $\lambda_k\left(\Omega-V_\epsilon\right) \geq \lambda_k(\Omega)$.

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