How to find the eigenvalue and eigenfunction of Laplacian?

Question

Define a bounded domain $\Omega=(0,a)\times(0,b)$ What is the eigenvalue and eigenfunction of the Laplacian with homogeneous boundary condition? my first thought is something like $sin(n\pi x/a)sin(n\pi y/b), cos(n\pi x/a)cos(n\pi y/b)$

Answer 1

You should have specified what the homogeneous boundary conditions are: Dirichlet, Neumann, Robin, or a mix of these? For the Dirichlet condition, it's indeed the product of sines; for Neumann, the product of cosines. However, the frequency need not be the same in both $x$ and $y$ variable: $$\phi_{m,n}(x,y) = \sin \frac{m \pi x}{a}\,\sin\frac{n \pi x}{b},\quad m,n\ge 1 $$ is the complete set of eigenvalues for the Dirichlet boundary condition. Similarly, $$\psi_{m,n}(x,y) = \cos \frac{m \pi x}{a}\,\cos\frac{n \pi x}{b},\quad m,n\ge 0 $$ are the Neumann eigenfunctions.

It is easy to check that the above are indeed eigenfunctions. The fact that there are no others follows from the fact that their linear span is dense in $L^2$. Sketch of proof, based on the corresponding 1-dimensional theorem:

• approximate an $L^2$ function on rectangle by continuous ones;
• approximate a continuous function by sums of characteristic functions of smaller rectangles $I\times J$ (where $I$ and $J$ are intervals).
• approximate $\chi_I$ and $\chi_J$ by appropriate trigonometric polynomials, and take the product of these polynomials.