Ebook: Analysis and Numerics of Partial Differential Equations

This volume is a selection of contributions offered by friends, collaborators, past students in memory of Enrico Magenes. The first part gives a wide historical perspective of Magenes' work in his 50-year mathematical career; the second part contains original research papers, and shows how ideas, methods, and techniques introduced by Magenes and his collaborators still have an impact on the current research in Mathematics. Read more... Analysis and Numerics of Partial Differential Equations; Preface; Contents; List of Contributors; Part I: A Historical Perspective; Personal Memories; Some Aspects of the Research of Enrico Magenes in Partial Differential Equations; 1 The Beginnings in Modena; 2 The Years in Genoa with G. Stampacchia; 3 The Collaboration with J.-L. Lions in the Study of Boundary Value Problems; References; Enrico Magenes and the Dam Problem; Inverse Problems in Electrocardiology; 1 Introduction; 2 Cardiac Body Surface Maps and Inverse Potential Problems. 3 Cardiac Excitation Sources Models and Inverse Excitation Wavefront ProblemsReferences; Stefan Problems and Numerical Analysis; 1 Stefan Problems and Semigroups: Analysis and Numerical Approximation; 2 Stefan Problems in a Concentrated Capacity; 3 Approximation of Interfaces, Adaptivity and Applications; References; Enrico Magenes and the Teaching of Mathematics; 1 Introduction; 2 Magenes and the Teaching of Mathematics: A Constant Commitment; 3 Magenes and the Research in Mathematics Education; 4 Magenes and the Research Unit in Pavia: The Project ""Mathematics as a Discovery"" 5 Two Talks as Keynote Speaker6 Conclusion; References; List of Mathematical Works Authored or Edited by Enrico Magenes; References; Part II: Recent Developments; Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below-The Compact Case; 1 Introduction; 2 Preliminary Notions; 2.1 Absolutely Continuous Curves and Slopes; 2.2 The Space (P(X), W2); 2.3 Geodesically Convex Functionals and Their Gradient Flows; Energy Dissipation Equality; Evolution Variational Inequality; 3 Hopf-Lax Formula and Hamilton-Jacobi Equation; 4 Weak Definitions of Gradient. 4.1 The ""Vertical"" Approach: Minimal Relaxed SlopeLaplacian: Definition and Basic Properties; 4.2 The ""Horizontal"" Approach: Weak Upper Gradients; Negligible Sets of Curves and Functions Sobolev Along a.e. Curve; A Bound from Below on Weak Gradients; 4.3 The Two Notions of Gradient Coincide; 4.4 Comparison with Previous Approaches; 5 The Relative Entropy and Its W2-Gradient Flow; 6 The Heat Flow as Gradient Flow; 7 A Metric Brenier Theorem; 8 More on Calculus on Compact CD(K, infty) Spaces; 8.1 On Horizontal and Vertical Derivatives Again; 8.2 Two Important Formulas. 9 Riemannian Ricci BoundsReferences; Spaces of Finite Element Differential Forms; 1 Introduction; 2 Preliminaries; 2.1 The Assembly of Finite Element Spaces; 2.2 The Lagrange Finite Element Family; 2.3 Exterior Calculus; 3 Families of Finite Element Differential Forms on Simplicial Meshes; 3.1 The Polynomial Space PrLambdak; 3.2 The Koszul Complex; 3.3 The Polynomial Space Pr-Lambdak; 3.4 The Pr-Lambdak(Th) Family of Finite Element Differential Forms; 3.5 The PrLambdak(Th) Family of Finite Element Differential Forms; 3.6 Historical Notes