# Tagged Questions

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### A domain in which the dirichlet laplacian has eigen values of all orders

I am trying to come up with an example to the following. Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., ...
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### On max-min representation for the principal eigenvalue of second order elliptic operator

(Just to be upfront about things, this is a homework problem.) I'm asked to show that the principal eigenvalue, $\lambda_1$ of an uniformly elliptic operator can be represented by ...
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### compare of eigenvalues $\lambda_1(D_a)$ and $\lambda_1(D_c)$.

Let $f(x)$ be a smooth function on $[-1,1]$, such that $f(x)>0$ for all $x\in(-1,1)$,$f(-1)=f(1)=0$. consider $\gamma\subset\Bbb{R}^2$ the graph of the $f(x)$. Let $T_a$ the symmetry with respect ...
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### the first eigenfunction of Dirichlet problem

Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
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### Stuck on Laplace's Equation

I am trying to solve the following: $-\theta_{yy}-\theta_{xx}=0$ $\theta(0,y)=-1$ $\theta(1,y)=1$ $\theta_y(x,0)=1$ $\theta_y(x,1)=0$ I can separate this into two different ...
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### Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I've already posted this question on Physics.SE, but I thougth it could be useful to ask also here. No problem if moderators will ask me to cancel this thread... But, please, have mercy! :-D Let ...
Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$: \lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...