Geometry of PDEs and related problems

Preface......Page 6
Contents......Page 10
1 Stable Solutions to Some Elliptic Problems: Minimal Cones, the Allen-Cahn Equation, and Blow-Up Solutions......Page 13
 1.1 Minimal Cones......Page 14
  1.1.1 The Simons Cone. Minimality......Page 16
  1.1.2 Hardy's Inequality......Page 26
  1.1.3 Proof of the Simons Lemma......Page 28
   1.1.4.1 Harmonic Maps......Page 33
   1.1.4.2 Free Boundary Problems......Page 35
   1.1.4.3 Nonlocal Minimal Surfaces......Page 36
 1.2 The Allen-Cahn Equation......Page 37
  1.2.1 Minimality of Monotone Solutions with Limits 1......Page 39
  1.2.2 A Conjecture of De Giorgi......Page 43
  1.2.3 The Saddle-Shaped Solution Vanishing on the SimonsCone......Page 45
 1.3 Blow-Up Problems......Page 47
  1.3.1 Stable and Extremal Solutions: A Singular Stable Solution for n ≥10......Page 48
  1.3.2 Regularity of Stable Solutions. The Allard and Michael-Simon Sobolev Inequality......Page 50
 Appendix: A Calibration Giving the Optimal Isoperimetric Inequality......Page 54
 References......Page 56
2 Isoperimetric Inequalities for Eigenvalues of the Laplacian......Page 58
  2.1.1 Notation and Sobolev Spaces......Page 59
   2.1.2.1 Abstract Spectral Theory......Page 60
   2.1.2.2 Application to the Laplacian......Page 61
  2.1.3 Properties of Eigenvalues......Page 62
   2.1.4.2 Disks......Page 63
  2.1.5 Min-Max Principles and Applications......Page 64
   2.1.5.2 Nodal Domains......Page 65
  2.1.6 Topological Derivative......Page 66
  2.2.1 The Faber-Krahn Inequality......Page 67
  2.2.2 A Quantitative Version of Faber-Krahn Inequality......Page 68
  2.2.3 The Case of Polygons......Page 69
   2.2.3.1 The Case of Triangles and Quadrilaterals......Page 74
   2.2.3.2 A Challenging Open Problem......Page 75
  2.2.4 Domains in a Box......Page 76
  2.2.5 Multi-Connected Domains......Page 80
   2.2.5.1 Optimizing with a Given Obstacle......Page 81
   2.2.5.2 Finding the Shape and the Location of the Obstacle......Page 83
  2.3.1 Minimizing λ2......Page 85
   2.3.2.1 Optimality Conditions......Page 87
   2.3.2.2 Regularity of the Optimal Domain......Page 88
  2.4.1 Existence......Page 89
  2.4.2 Connectedness of Minimizers......Page 90
   2.4.3.1 Perimeter Constraint......Page 93
   2.4.3.2 Diameter Constraint......Page 96
 References......Page 97
 3.1 Introduction: Emergence of Topological Structures in Elliptic  PDEs......Page 100
 3.2 Critical Points of Green's Functions on Complete Manifolds......Page 103
  3.2.1 Li-Tam Green's Functions......Page 104
  3.2.2 A Topological Upper Bound on Surfaces......Page 105
  3.2.3 Critical Points in Higher Dimensions......Page 110
 3.3 General Strategy and Two Technical Tools: Thom's Isotopy Theorem and a Runge-Type Global Approximation Theorem......Page 112
  3.3.1 Thom's Isotopy Theorem......Page 113
  3.3.2 A Runge-Type Global Approximation Theorem for the Helmholtz Equation with Optimal Decay at Infinity......Page 114
 3.4 Monochromatic Waves: Nodal Sets of Solutions to the Helmholtz Equation......Page 119
 3.5 Emergence of Knotted Structures in High-Energy Eigenfunctions: Berry's Conjecture......Page 121
 3.6 The Linear Regime of Nonlinear Equations: Nodal Sets of the Allen–Cahn Equation......Page 127
 References......Page 129
4 Symmetry Properties for Solutions of Higher-Order Elliptic Boundary Value Problems......Page 131
 4.1 Linear Problems: Weak Solutions, Eigenvalues, Regularity, Green Functions......Page 132
 4.2 Symmetry, Simplicity and Positivity for First Eigenfunctions......Page 135
 4.3 Symmetry for Nonlinear Problems by Uniqueness and Non-resonance......Page 138
 4.4 Symmetry via Moving Plane Method: An Example from Potential Theory......Page 142
  4.4.1 Asymptotic Expansion and the Role of the Barycenter......Page 146
  4.4.2 The Moving Plane Method......Page 147
 4.5 An Example of Symmetry in an Overdetermined 4th Order Problem......Page 151
 References......Page 154
  5.1.1 Background......Page 156
  5.1.2 Overview of the Content......Page 157
  5.2.2 Hele-Shaw Flow......Page 158
  5.2.3 An Optimal Stopping Problem......Page 159
  5.2.4 Modeling Financial Derivatives......Page 160
  5.2.5 Smash Sums and Internal DLA......Page 161
  5.3.1 Existence, and Uniqueness......Page 162
  5.3.2 Optimal Regularity and Non-degeneracy of Solutions......Page 166
  5.3.3 Regularity of FB: Local and Global Analysis......Page 170
  5.4.1 Bernoulli Type FB......Page 176
  5.4.2 Broken PDEs with FB......Page 182
   5.4.2.1 Quasi-Linear Case: Relation to Composites......Page 183
   5.4.2.3 Local Analysis and Regularity......Page 184
   5.4.2.5 Proof of Smoothness of Nodal Sets......Page 186
  5.4.3 Non-variational Problems......Page 187
   5.4.3.1 Regularity of Potentials (Heuristics)......Page 188
   5.4.3.2 Proof of John Andersson's Dichotomy......Page 190
   5.4.4.1 Thin Obstacles: Semipermeable Membranes......Page 191
   5.4.4.2 Nonlocal Problems......Page 192
   5.4.4.3 Local Analysis of Fractional Obstacle Problem......Page 194
  5.5.1 Optimal Switching......Page 196
  5.5.2 Minimization Problems......Page 198
   5.5.2.2 Case f(u) = χ{|u| >0 }: Thermal Insulation......Page 199
   5.5.2.3 Properties of Solutions......Page 200
 References......Page 205