内容

目次

Part I. Introduction: 1. The isoperimetric problem; 2. The isoperimetricinequality in the plane; 3. Preliminaries; 4. Bibliographic notes; Part II.Differential Geometric Methods: 1. The C2 uniqueness theory; 2. The C1isoperimetric inequality; 3. Bibliographic notes; Part III. Minkowski Areaand Perimeter: 1. The Hausdorff metric on compacta; 2. Minkowski area andSteiner symmetrization; 3. Application: the Faber-Krahn inequality; 4.Perimeter; 5. Bibliographic notes; Part IV. Hausdorff Measure and Perimeter:1. Hausdorff measure; 2. The area formula for Lipschitz maps; 3.Bibliographic notes; Part V. Isoperimetric Constants: 1. Riemannian geometricpreliminaries; 2. Isoperimetric constants; 3. Discretizations andisoperimetric inequalities; 4. Bibliographic notes; Part VI. AnalyticIsoperimetric Inequalities: 1. L2-Sobolev inequalities; 2. The compact case;3. Faber-Kahn inequalities; 4. The Federer-Fleming theorem: the discretecase; 5. Sobolev inequalities and discretizations; 6. Bibliographic notes;Part VII. Laplace and Heat Operators: 1. Self-adjoint operators and theirsemigroups; 2. The Laplacian; 3. The heat equation and its kernels; 4. Theaction of the heat semigroup; 5. Simplest examples; 6. Bibliographic notes;Part VIII. Large-Time Heat Diffusion: 1. The main problem; 2. The Nashapproach; 3. The Varopoulos approach; 4. Coulhon's modified Sobolevinequality; 5. The denoument: geometric applications; 6. Epilogue: theFaber-Kahn method; 7. Bibliographic notes; Bibliography.