Singular limits of sign-changing weighted eigenproblems

Article type: Research Article

Authors: Kielty, Derek^{; *}

Affiliations: Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.. E-mail: dkielty2@illinois.edu

Correspondence: [*] Corresponding author. E-mail: dkielty2@illinois.edu.

Abstract: 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.

Keywords: Spectral theory, indefinite, singular limits, mixed boundary conditions, Dirichlet, Neumann, Laplacian, bi-Laplacian, dynamical boundary conditions

DOI: 10.3233/ASY-201615

Journal: Asymptotic Analysis, vol. 122, no. 1-2, pp. 165-200, 2021