Let $H$ be the Hilbert space of square integrable functions defined on a 2-dimension real domain $S \subseteq \mathbf{R}^2$. I am interested in studying the eigenstructure of the Laplacian operator defined for complex-valued functions that vanish on the boundary of $S$.
For instance, consider the domain $S$ defined by the pseudo-square of unit length edges, where one of the edges is replaced by a sinusoidal function. In particular, the boundary of this square is defined by the edges: $(0,y), 0\leq y \leq 1$; $(1,y), 0\leq y \leq 1$; $(x,1), 0\leq x \leq 1$; and $(x,a\, \mathrm{sin}(2 \pi x)), 0\leq x \leq 1$ and $0 < a < 1$. Therefore, I am investigating the self-adjoint non-negative eigenvalue problem $$ -(u_{xx} + u_{yy}) = \lambda u \tag{1} $$ for complex-valued functions $u(x,y)$ such that $u(x,y) = 0$ for each $(x,y) \in S$.
My questions are:
1) does this problem have non-trivial eigenfunctions? 2) is it possible to represent the non-trivial eigenfunctions analytically?
Thankx in advance.
The Laplace operator with Dirichlet boundary conditions on a bounded domain is self-adjoint with compact resolvent: see e.g. Laplace operator - Spectral theory. Thus it does have eigenfunctions, in fact a complete orthonormal set of them. But it is unlikely that you'll be able to express them in closed form.