The spectrum of the Dirichlet Laplacian

Abstract

We study the Dirichlet eigenvalue problem Δu =λu in Ω, u|_dΩ=0 for open and bounded sets Ω in R^n. An analogous distributional problem is formulated using the Sobolev space H^1_0(Ω). Viewing the Laplacian as an operator on this space we construct a compact, symmetric, positive-definite, and bounded inverse which allows us to use classical results from functional analysis to characterize its spectrum. This allows us to see that the eigenvalues of the distributional problem can be arranged in a positive, increasing sequence that only accumulates at infinity. Regularity theorems involving elliptic linear partial differential operators are proven to relate the eigenfunctions of the distributional problem to the classical Dirichlet eigenfunctions. We prove that if the boundary of Ω is C^k for k an integer satisfying 2k>n, then the eigenfunction of the distributional problem are Dirichlet eigenfunctions and are smooth in Ω and continuous up to the boundary. Existence and uniqueness theorems are established concerning the classical Dirichlet boundary value problem for such Ω. In particular, we establish existence and uniqueness of Green's functions. In the cases n=2 and n=3 we establish an equivalence between the eigenfunctions of the distributional problem and the Dirichlet eigenfunctions assuming the boundary of Ω is C^2.