[EX] First Eigenvalue of the Neumann Laplacian
Inviato: 09/04/2013, 23:33
Exercise:
1. Let \(\mathcal{R}(a,b)\subseteq \mathbb{R}^2\) be any open rectangle with sides of length $a,b>0$.
Using the separation of variables find the first nonzero eigenvalue \(\mu_2(a,b)\) of the Neumann Laplacian in \(\mathcal{R}(a,b)\), i.e. find the smallest $\mu >0$ s.t. the problem:
\[ \tag{N}
\begin{cases} -\Delta u= \mu\ u &\text{, in } \mathcal{R}(a,b)\\
\frac{\partial u}{\partial \nu} =0 &\text{, on } \partial \mathcal{R}(a,b)
\end{cases}
\]
(where $\frac{\partial u}{\partial \nu}$ denotes the outward normal derivative) has some nonconstant solutions with separate variables.*
2. Are there values of $a,b>0$ which minimize/maximize $\mu_2(a,b)$ under measure constraint, i.e. under condition \(| \mathcal{R}(a,b) |=ab=\text{constant}\)?
__________
* I used $\mu_2$ because, actually, the first eigenvalue of the Neumann Laplacian is always $\mu_1=0$. In fact, every constant function $u(x,y)=c$ is a solution of the eigenvalue problem (N) associated with the $0$ eigenvalue and it can be easily proved that every other eigenvalue of (N) is positive.
1. Let \(\mathcal{R}(a,b)\subseteq \mathbb{R}^2\) be any open rectangle with sides of length $a,b>0$.
Using the separation of variables find the first nonzero eigenvalue \(\mu_2(a,b)\) of the Neumann Laplacian in \(\mathcal{R}(a,b)\), i.e. find the smallest $\mu >0$ s.t. the problem:
\[ \tag{N}
\begin{cases} -\Delta u= \mu\ u &\text{, in } \mathcal{R}(a,b)\\
\frac{\partial u}{\partial \nu} =0 &\text{, on } \partial \mathcal{R}(a,b)
\end{cases}
\]
(where $\frac{\partial u}{\partial \nu}$ denotes the outward normal derivative) has some nonconstant solutions with separate variables.*
2. Are there values of $a,b>0$ which minimize/maximize $\mu_2(a,b)$ under measure constraint, i.e. under condition \(| \mathcal{R}(a,b) |=ab=\text{constant}\)?
__________
* I used $\mu_2$ because, actually, the first eigenvalue of the Neumann Laplacian is always $\mu_1=0$. In fact, every constant function $u(x,y)=c$ is a solution of the eigenvalue problem (N) associated with the $0$ eigenvalue and it can be easily proved that every other eigenvalue of (N) is positive.