Isoperimetric inequalities and applications

Preface ix
I Geometrical inequalities 1
1 Geometrical inequalities in the Euclidean space 1
1.1 Introduction 1
1.2 The Payne-Weinberger inequality 2
1.3 Parallel sets 3
1.4 Parallel curves in the plane 4
2 On a class of isoperimetric inequalities in the plane 10
2.1 Definitions 10
2.2 Inequalities for sectors 10
2.3 Rotation of a curve 14
2.4 Main results 17
3 Isoperimetric inequalities on surfaces 19
3.1 Introduction 19
3.2 Surfaces with constant Gaussian curvature 26
3.3 Total curvature 29
3.4 The pasting together of surfaces 32
3.5 Isoperimetric inequalities 35
3.6 Proofs 38
3.7 Consequences and complements 40
3.8 Mean value theorems 43
II Symmetrizations 47
1 The Schwarz symmetrization 47
1.1 Definitions and basic properties 47
1.2 Inequalities related to symmetrizations 52
2 Applications of the Schwarz symmetrization 56
2.1 Electrostatic capacity 56
2.2 The Green’s function 58
2.3 Torsional rigidity 63
2.4 Poisson problems 67
3 The a-symmetrization 73
3.1 Definitions 73
3.2 Poisson problems with mixed boundary conditions 75
4 Symmetrizations on surfaces 79
4.1 Generalized Schwarz symmetrization 79
4.2 Applications 82
4.3 Generalized a-symmetrization 88
III Eigenvalue problems 91
1 Introduction 91
1.1 Examples of eigenvalue problems 91
1.2 Abstract eigenvalue problems 96
1.3 Variational characterization of eigenvalues for special problems 100
2 Isoperimetric inequalities for eigenvalue problems 104
2.1 Lower bounds for the fundamental frequency of a membrane 104
2.2 Lower bounds for the first eigenvalue of the Beltrami operator 106
2.3 The Payne-Weinberger inequality for wedge-like membranes 108
2.4 A Faber-Krahn inequality for a general second-order operator 110
2.5 Membranes with mixed boundary conditions 113
3 Conformal transplantation 116
3.1 Preliminary results 116
3.2 Inequalities for the sums of reciprocal eigenvalues of fixed membranes 119
3.3 Free membranes 121
3.4 On quadrilaterals, trilaterals and their eigenvalues 128
3.5 Inequalities for the Stekloff eigenvalues 135
4 Prescribed level surfaces 141
4.1 Harmonic transplantation 141
4.2 The method of parallel level lines 143
4.3 Applications of formula (3.36) 144
4.4 Similar level lines 150
5 Two isoperimetric inequalities for the N-dimensional free membrane problem 152
6 The Cheeger inequality and some supplements on the method of symmetrization 158
7 The Hopf maximum principle and an isoperimetric inequality for the Stekloff problem 161
8 The Barta inequalities 163
IV Boundary- and initial-value problems 165
1 Linear elliptic boundary-value problems 165
1.1 Introduction 165
1.2 Comparison theorems 166
1.3 Remarks on generalized concavity and convexity 171
1.4 A differential inequality 174
2 Non-linear elliptic boundary-value problems 184
2.1 Introduction 184
2.2 Numerical bounds for 2* 186
2.3 Linearized problem; further properties of minimal and non-minimal solutions 188
2.4 Extremal properties of the sphere 193
2.5 On the problem Au + 2e“ = 0 196
2.6 Proofs of geometrical isoperimetric inequalities 203
3 Parabolic equations 206
3.1 Introduction 206
3.2 Special case 209
3.3 Symmetrizations of families of functions 211
3.4 Isoperimetric inequalities for linear parabolic problems 213
3.5 Non-linear parabolic equations 216
References 220
Index
221