Geometric Properties for Parabolic and Elliptic PDE's

Cover......Page 1
Geometric Properties for Parabolic and Elliptic PDE\'s......Page 3
Preface......Page 5
Contents......Page 7
1 Introduction......Page 9
2 Group Invariance of Weak Solutions for Parabolic Problems......Page 11
3 Stability Analysis of Group Invariant Asymptotic Profiles......Page 15
3.1 Asymptotic Profiles for Fast Diffusion Equations......Page 16
3.2 Stability and Instability of G-Invariant Profiles......Page 18
3.3 Proof of Theorem 3......Page 19
3.4 Proof of Theorem 4......Page 21
3.5 Applications of Stability Criteria......Page 22
References......Page 23
1 Introduction......Page 25
2 Preliminaries......Page 28
3 Hardy-Rellich Type Inequalities with a Boundary Term......Page 32
4 Proof of Theorem 1 and Corollary 1......Page 34
5 Radial Setting......Page 37
Appendix......Page 38
References......Page 40
1 Introduction......Page 42
2 Preliminaries......Page 43
3 The Main Result......Page 45
4 The p-Concave Envelope of a Function......Page 46
5 Proof of Theorem 1......Page 49
6 Examples......Page 52
References......Page 54
1 Introduction......Page 56
2.1 Properties of the Maximal Folding Function......Page 58
3 Relevant Points Contained in the Heart......Page 60
3.1 An Estimate of the Heart\'s Diameter......Page 61
3.2 The p-Moments of K and More......Page 63
3.3 On Fraenkel\'s Asymmetry......Page 67
4 Estimating the Volume of the Heart......Page 69
References......Page 72
1 Introduction......Page 74
2 Preliminary Results......Page 75
3 Viscosity Euler-Lagrange Equation......Page 80
References......Page 89
1 Introduction......Page 91
2 Preliminary Results: The Weierstrass Method......Page 94
3 Kato\'s Inequality: An Alternative Proof......Page 96
4 Sketch of the Proof of Theorem 2......Page 100
References......Page 101
1 Introduction......Page 103
2 Results......Page 106
3 Geometric Preliminaries......Page 109
4.1 Proof of Theorem 2......Page 112
4.2 Proof of Theorem 1......Page 117
4.3 Proof of Theorem 3......Page 118
5 Some Open Problems......Page 119
References......Page 120
1 The Fractional Laplacian Operator......Page 122
2.2 A Conjecture of De Giorgi......Page 124
3.1 The Symmetry Problem for (-Delta)s......Page 125
3.2 Gamma-Convergence for (-Delta)s......Page 126
3.3 Density Estimates for the Level Sets......Page 128
3.4 Few Words on the Nonlocal Perimeter......Page 129
3.5 Singularities of Nonlocal Minimal Surfaces......Page 132
References......Page 133
1 Introduction......Page 136
2 Convex Envelope......Page 139
3 Proofs of the Main Theorems......Page 142
Appendix:  The Cycloid......Page 144
References......Page 147
1 Introduction......Page 148
2 Preliminaries......Page 152
3.1 Solvability of (4)......Page 157
3.2 Achievement of d1(epsilon,alpha1,alpha2) and Proof of Theorem 1......Page 159
4 Proof of Theorem 2......Page 165
5 Proof of Theorem 3......Page 166
References......Page 168
1 Introduction......Page 170
2 Preliminaries......Page 173
2.2 Preliminary Results......Page 174
3 Proof of Theorem 2......Page 176
References......Page 182
1 Introduction......Page 184
Existence of Eigenvalues......Page 186
Formulation of the Domain Variation and Eigenvalue Problem......Page 187
2 Transformation of the Problem......Page 188
Justification......Page 191
4 Analysis of the Perturbation of the Electromagnetic Eigenvalue......Page 193
Variational Equation......Page 194
Evaluation of A, B......Page 198
5 Justification of the Formula......Page 199
References......Page 204
1 Introduction......Page 205
2 Derivation of the Limiting Equation......Page 207
3 Main Results......Page 209
3.1 Solution Set of (14)......Page 210
3.2 Linearized Equations......Page 211
4.1 Proofs of Theorems 2 and 3......Page 213
4.2 Proof of Theorem 1......Page 219
Appendix......Page 224
References......Page 225
1 Introduction......Page 227
2 A Priori Estimates......Page 229
3 Comparison Principle......Page 235
References......Page 237
1 The Helfrich Variational Problem and Its Gradient Flow......Page 240
2 Proof of Theorem 1......Page 244
3.1 Sketch of the Proof of Theorem 2 (i)......Page 245
3.2 Proof of Theorem 2 (ii)......Page 247
4 On the Uniqueness for the Degenerate Case......Page 250
5 Open Problems......Page 251
References......Page 252
1 Introduction......Page 254
2 Main Results......Page 256
3 Mountain Pass Geometry......Page 258
4 Preliminary Results......Page 259
5 Estimate of the Mountain Pass Level c......Page 263
6 The Infimum A Is Attained......Page 266
References......Page 269
Stationary Level Surfaces and Liouville-Type Theorems Characterizing Hyperplanes......Page 271
1 Introduction......Page 272
2 Proof of Theorem 1......Page 276
3 Proof of Theorem 2......Page 277
4 Proof of Theorem 3......Page 281
5 Concluding Remarks......Page 283
References......Page 284
1 Introduction......Page 285
2 New Pohozaev Identity for the Green Function......Page 289
3 Proof of Theorem 1......Page 292
References......Page 293