1996/01/01 by Michel L. Lapidus, Carl Pomerance · 2 citations
Mathematics · Computer Science · #Spectral Theory in Mathematical Physics #Analytic and geometric function theory #Advanced Mathematical Modeling in Engineering #Mathematics #Hypersurface #Lebesgue measure #Eigenvalues and eigenvectors #Laplace operator #Null set #Combinatorics #Hausdorff dimension #Pure mathematics #Mathematical analysis #Lebesgue integration #Set (abstract data type) #Physics
paper · doi:10.1017/s0305004100074053
openalex publication_date 1996/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
Let Ω be a non-empty open set in ℝ n with finite ‘volume’ ( n -dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions): where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H 1 (Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ 1 ≤ λ 2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let