2 Basic concepts

Chapters in this book

1. Frontmatter I
2. Preface V
3. Contents VII
4. 1 Overview of the basic problems 1
5. 2 Basic concepts 7
6. 3 Spherical harmonics and eigenvalue problems 38
7. 4 Variational formulas 52
8. 5 Geometric inequalities, convolutions, cost functions 71
9. 6 Domain variations for energies 99
10. 7 Discussion of the main results 119
11. 8 General strategy and applications 131
12. 9 Eigenvalue problems 148
13. 10 Quantitative estimates 167
14. 11 The Robin eigenvalues for α < 0 173
15. 12 Problems with infinitely many positive and negative eigenvalues 188
16. 13 The torsion problem for α < 0 195
17. 14 Problems in annular domains 206
18. 15 The first buckling eigenvalue of a clamped plate 217
19. 16 A fourth order Steklov problem 236
20. A General remarks 253
21. B Geometry 257
22. C Sobolev spaces and inequalities 263
23. D Bilinear forms 265
24. Notation 269
25. Bibliography 271
26. Index 277