Shape Optimization: Variations of Domains and Applications

This Book Investigates How Domain Dependent Quantities From Geometry And Physics Behave When The Domain Is Perturbed. Of Particular Interest Are Volume- And Perimeter-preserving Perturbations. The First And Second Derivatives With Respect To The Perturbation Are Exploited For Domain Functionals Like Eigenvalues, Energies And Geometrical Quantities. They Provide Necessary Conditions For Optimal Domains And Are Useful When Global Approaches Like Symmetrizations Fail. The Book Is Exampledriven And Illustrates The Usefulness Of Domain Variations In Various Applications. Frontmatter -- Preface -- Contents -- 1 Overview Of The Basic Problems -- 2 Basic Concepts -- 3 Spherical Harmonics And Eigenvalue Problems -- 4 Variational Formulas -- 5 Geometric Inequalities, Convolutions, Cost Functions -- 6 Domain Variations For Energies -- 7 Discussion Of The Main Results -- 8 General Strategy And Applications -- 9 Eigenvalue Problems -- 10 Quantitative Estimates -- 11 The Robin Eigenvalues For α < 0 -- 12 Problems With Infinitely Many Positive And Negative Eigenvalues -- 13 The Torsion Problem For α < 0 -- 14 Problems In Annular Domains -- 15 The First Buckling Eigenvalue Of A Clamped Plate -- 16 A Fourth Order Steklov Problem -- A General Remarks -- B Geometry -- C Sobolev Spaces And Inequalities -- D Bilinear Forms -- Notation -- Bibliography -- Index Catherine Bandle, Alfred Wagner. Issued Also In Print. Mode Of Access: Internet Via World Wide Web. In English.