Dirichlet Laplacian ($-{\Delta }_D$)

The nonnegative Dirichlet Laplacian is the self-adjoint realization of $-\mathrm{div}\mathrm{grad}$ with zero boundary values. On a bounded domain, its weak form is the Dirichlet energy on $H_0^1$. For polygonal domains the weak operator domain is preferable to a requirement of smoothness at corners.

On $0<x<L_x$, $0<y<L_y$ with positive side lengths and zero Dirichlet boundary conditions, separation of variables for $-\Delta u=\lambda u$ giveswith the displayed eigenvalues. The one-dimensional endpoint conditions exclude constant and hyperbolic factors and force the positive integer sine frequencies. The product Fourier sine series basis is complete in the square-integrable functions on the rectangle. Consequently an eigenfunction with prescribed $\lambda$ is a linear combination of exactly those modes with ${\lambda }_{mn}=\lambda$; if none match, only the zero solution exists. Distinct pairs with the same sum give an eigenvalue with geometric multiplicity greater than one. On a square of side $\pi$, for instance, $\lambda =25$ has precisely the pairs $(3,4)$ and $(4,3)$, so its eigenspace has dimension two.

If ${\beta }_1,{\beta }_2,{\beta }_3$ are the affine barycentric coordinates of an equilateral triangle of side $\ell$, then $h={\prod }_j\sin (\pi {\beta }_j)=\frac{1}{4}{\sum }_j\sin (2\pi {\beta }_j)$ vanishes on the boundary and satisfies $\Delta h=-16{\pi }^{2}h/(3{\ell }^2)$. Thus the problem $\Delta q-4\lambda q=0$ with zero Dirichlet boundary data has distinct solutions zero and $h$ at the displayed negative parameter. Their normal traces differ, demonstrating failure of a uniquely determined Dirichlet-to-Neumann map at resonance.

For the positive Dirichlet Laplacian $A$ on a bounded smooth domain, with normalized eigenfunctions $e_j$ and positive eigenvalues ${\mu }_j$, define $A^{\alpha }u={\sum }_j{\mu }_j^{\alpha }(u,e_j)e_j$ on the vectors for which ${\sum }_j{\mu }_j^{2\alpha }|(u,e_j){|}^2<\infty$. The resulting Hilbert space embeds in the Sobolev space $H^{2\alpha }$, with the boundary conditions appropriate to the exponent. In particular $H_{\alpha }$ embeds in $C^{0,\beta }$ whenever $0<\beta <\min (1,2\alpha -n/2)$. High powers impose further boundary compatibility; ordinary smoothness alone does not imply membership in every fractional domain.