What is the fundamental different between a Neumann eigenvalue problem and Dirichlet eigenvalue problem? I know that for DEP, we just fix the boundary (e.g. a drum), but what about the NEP. Now, consider a rectangle $\Omega = [0,l] \times [0,m]$. Separate variables using cartesian coordinates $x$ and $y$. That is, look for solution of the form $\varphi(x,y)=f(x)g(y)$
- Dirichlet boundary condition $\varphi| \partial \Omega=0$.
The eigenfunction are $$\varphi_{j,k}(x,y)= \sin(\frac{j \pi}{l}x) \sin (\frac{k \pi}{m}y) \text{ for } j,k \geq 1$$ and have eigenvalues $$\lambda_{j,k} = (\frac{j \pi}{l})^2 + (\frac{k \pi}{m})^2 \text{ for } j,k \geq 1$$
- Neumann boundary conditions $\partial_{\nu} \varphi | \partial \Omega = 0 $
The eigenfunction are $$\varphi_{j,k}(x,y)= \cos(\frac{j \pi}{l}x) \cos (\frac{k \pi}{m}y) \text{ for } j,k \geq 0$$ and have eigenvalues $$\lambda_{j,k} = (\frac{j \pi}{l})^2 + (\frac{k \pi}{m})^2 \text{ for } j,k \geq 0$$
With these informations, does someone could explain to me the difference between NEP and DEP? I would like fundamentally how behave the boundary of NEP.