Eigencurves for linear elliptic equations

¹ Department of Mathematics, North Carolina Agricultural and Technical State University, 232 Marteena Hall, 1601 East Market Street, Greensboro, NC 27411, USA.
² Department of Mathematics and Statistics, Wake Forest University, PO Box 7388, 127 Manchester Hall, Winston-Salem, NC 27109, USA.

^* Corresponding author: marivas@ncat.edu

Received: 27 May 2017
Accepted: 18 June 2018

Abstract

This paper provides results for variational eigencurves associated with self-adjoint linear elliptic boundary value problems. The elliptic problems are treated as a general two-parameter eigenproblem for a triple (a, b, m) of continuous symmetric bilinear forms on a real separable Hilbert space V . Geometric characterizations of eigencurves associated with (a, b, m) are obtained and are based on their variational characterizations described here. Continuity, differentiability, as well as asymptotic, results for these eigencurves are proved. Finally, two-parameter Robin–Steklov eigenproblems are treated to illustrate the theory.