On Shape Optimizing the Ratio of the First Two Eigenvalues of the Laplacian (Classic Reprint) - Hardcover

Jean-Pierre A. Haeberly

Hardcover

ISBN 10: 0267869975 ISBN 13: 9780267869978

Publisher: Forgotten Books, 2022

Synopsis

Unlock the math behind shaping spaces to push the smallest Laplacian eigenvalues apart.

This nonfiction work explores a classic conjecture about how geometry controls the ratio of the first two eigenvalues, and it shows how researchers turn that question into a concrete optimization problem you can follow step by step.

The book frames the challenge as a numerical pursuit. By discretizing shapes with finite elements and using specialized algorithms, the author demonstrates how to balance precision with computation while tracking how eigenvalues respond to shape changes. It also explains how tools like generalized gradients and matrix pencils help compute descent directions in a stable, explainable way.

• Learn how to model perturbations of a disk and approximate the Laplacian’s behavior on complex shapes.
• See how the generalized gradient and optimization theory guide the search for shape changes that improve the target ratio.
• Discover how established algorithms are extended to handle non-smooth points where eigenvalues collide.
• Get insight into how numerical experiments are designed, run, and interpreted in this context.

Ideal for readers of applied mathematics, numerical analysis, and optimization who want a practical look at eigenvalue problems and shape design.

Edition details

• Publisher: Forgotten Books
• Publication date: 2022
• Language: English
• ISBN 10: 0267869975
• ISBN 13: 9780267869978
• Binding: Hardcover
• Number of pages: 77