# Eigencurves for linear elliptic equations

## TL;DR

This paper studies eigencurves of self-adjoint linear elliptic boundary value problems, providing variational characterizations, orthogonality properties, and asymptotic behaviors to understand their geometric structure.

## Contribution

It introduces a general framework for analyzing eigencurves via a two-parameter eigenproblem and establishes key properties like continuity, differentiability, and asymptotics.

## Key findings

- Variational characterizations of eigencurves
- Orthogonality results for eigenspaces
- Continuity and asymptotic behavior of eigencurves

## Abstract

This paper provides results for 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. Variational characterizations of the eigencurves associated with (a, b, m) are given and various orthogonality results for corresponding eigenspaces are found. Continuity and differentiability, as well as asymptotic results, for these eigencurves are proved. These results are then used to provide a geometric description of the eigencurves.