Bandle K. Isoperimetric inequalities and applications

New York: Pitman Publishing, 1980. - 235 p.Geometrical inequalities
Geometrical inequalities in the Euclidean spaceThe Payne-Weinberger inequality
Parallel sets
Parallel curves in the plane
On a class of isoperimetric inequalities in the plane
Definitions
Inequalities for sectors
Rotation of a curve
Main results
Isoperimetric inequalities on surfacesSurfaces with constant Gaussian curvature
Total curvature
The pasting together of surfaces
Isoperimetric inequalities
Proofs
Consequences and complements
Mean value theorems
Symmetrizations
The Schwarz symmetrization
Definitions and basic properties
Inequalities related to symmetrizations
Applications of the Schwarz symmetrization
Electrostatic capacity
The Green’s function
Torsional rigidity
Poisson problems
The a-symmetrization
Definitions
Poisson problems with mixed boundary conditions
Symmetrizations on surfaces
Generalized Schwarz symmetrization
Applications
Generalized a-symmetrization
Eigenvalue problemsExamples of eigenvalue problems
Abstract eigenvalue problems
Variational characterization of eigenvalues for special problems
Isoperimetric inequalities for eigenvalue problems
Lower bounds for the fundamental frequency of a membrane
Lower bounds for the first eigenvalue of the Beltrami operator
The Payne-Weinberger inequality for wedge-like membranes
A Faber-Krahn inequality for a general second-order operator
Membranes with mixed boundary conditions
Conformal transplantation
Preliminary results
Inequalities for the sums of reciprocal eigenvalues of fixed membranes
Free membranes
On quadrilaterals, trilaterals and their eigenvalues
Inequalities for the Stekloff eigenvalues
Prescribed level surfaces
Harmonic transplantation
The method of parallel level lines
Applications of formula (3.36)
Similar level lines
Two isoperimetric inequalities for the N-dimensional free membrane problem
The Cheeger inequality and some supplements on the method of symmetrization
The Hopf maximum principle and an isoperimetric inequality for the Stekloff problem
The Barta inequalities
Boundary- and initial-value problems
Linear elliptic boundary-value problemsComparison theorems
Remarks on generalized concavity and convexity
A differential inequality
Non-linear elliptic boundary-value problemsNumerical bounds for 2*
Linearized problem; further properties of minimal and non-minimal solutions
Extremal properties of the sphere
On the problem Au + 2e“ = 0
Proofs of geometrical isoperimetric inequalities
Parabolic equationsSpecial case
Symmetrizations of families of functions
Isoperimetric inequalities for linear parabolic problems
Non-linear parabolic equations