2018/12/31 by Derek Kielty · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Bilinear form #Bilinear interpolation #Boundary value problem #Differential Equations and Numerical Methods #Dirichlet boundary condition #Dirichlet eigenvalue #Dirichlet's principle #Eigenvalues and eigenvectors #Hilbert space #Limiting #Mathematical analysis #Mathematics #Neumann boundary condition #Nonlinear Partial Differential Equations #Operator (biology) #Sign (mathematics) #math.AP #math.SP #msc:35P15 #msc:47A07
paper · pdf · doi:10.3233/asy-201615
published in Asymptotic Analysis 122(1-2), 165-200 (IOS Press) · 34 pages, 3 figures, 2 tables
openalex publication_date 2020/05/13 · arxiv created 2020/11/12 · arxiv updated 2020/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. Our main theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We apply our results to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type boundary conditions, and a problem where the eigenvalue appears in both the equation and the boundary condition.