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