The Generalized Gradient at a Multiple Eigenvalue

✍ Scribed by S.J. Cox

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 329 KB
• Volume: 133
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1995.1117

✦ Synopsis

When a symmetric, positive, isomorphism between a reflexive Banach space (that is densely and compactly embedded in a Hilbert space) and its dual varies smoothly over a Banach space, its eigenvalues vary in a Lipschitz manner. We calculate the generalized gradient of the extreme eigenvalues at an arbitrary crossing. We apply this to the generalized gradient, with respect to a coefficient in an elliptic operator, of (i) the gap between the operator's first two eigenvalues and (ii) the distance from a prescribed value to the spectrum of the operator. "1995 Academic Press. Inc.