Stable Solutions of Elliptic Partial Differential Equations

Contents......Page 8
Preface......Page 12
1.1.1 Potential wells......Page 16
1.1.2 Examples of stable solutions......Page 20
1.2.1 Principal eigenvalue of the linearized operator......Page 24
1.2.2 New examples of stable solutions......Page 26
1.3.1 Uniqueness......Page 30
1.3.2 Nonuniqueness......Page 31
1.3.3 Symmetry......Page 33
1.4 Dynamical stability......Page 35
1.5 Stability outside a compact set......Page 39
1.6 Resolving an ambiguity......Page 41
2.1 Motivation......Page 44
2.2 Dimension N = 1......Page 45
2.3 Dimension N = 2......Page 49
2.4 Dimension N ≥ 3......Page 50
2.4.1 Stability analysis......Page 54
2.5 Summary......Page 59
3. Extremal solutions......Page 62
3.1.1 Defining weak solutions......Page 63
3.2.1 Uniqueness of stable weak solutions......Page 66
3.2.2 Approximation of stable weak solutions......Page 68
3.3 The stable branch......Page 73
3.3.1 When is λ* finite?......Page 76
3.3.2 What happens at λ = λ*?......Page 77
3.3.3 Is the stable branch a (smooth) curve?......Page 84
3.3.4 Is the extremal solution bounded?......Page 88
4.1 The radial case......Page 90
4.2 Back to the Gelfand problem......Page 95
4.3 Dimensions N = 1, 2, 3......Page 97
4.4 A geometric Poincaré formula......Page 100
4.5.1 Interior estimates......Page 103
4.5.2 Boundary estimates......Page 109
4.5.3 Proof of Theorem 4.5.1 and Corollary 4.5.1......Page 110
4.6 Regularity of solutions of bounded Morse index......Page 111
5.1 The Gelfand problem in the perturbed ball......Page 114
5.2 Flat domains......Page 125
5.3 Partial regularity of stable solutions in higher dimensions......Page 130
5.3.1 Approximation of singular stable solutions......Page 131
5.3.2 Elliptic regularity in Morrey spaces......Page 134
5.3.3 Measuring singular sets......Page 138
5.3.4 A monotonicity formula......Page 140
5.3.5 Proof of Theorem 5.3.1......Page 145
6.1 Classifying radial stable entire solutions......Page 152
6.2.1 The Liouville equation......Page 156
6.2.2 Dimension N = 2......Page 158
6.2.3 Dimensions N = 3, 4......Page 160
6.3.1 The critical case......Page 162
6.3.2 The supercritical range......Page 169
6.3.3 Flat nonlinearities......Page 173
7.1 Statement of the conjecture......Page 178
7.2.1 Phase transition phenomena......Page 179
7.2.2 Monotone solutions and global minimizers......Page 181
7.2.3 From Bernstein to De Giorgi......Page 187
7.3 Dimension N = 2......Page 188
7.4 Dimension N = 3......Page 189
8.1.1 The half-space......Page 194
8.1.2 Domains with controlled volume growth......Page 196
8.1.3 Exterior domains......Page 198
8.2.1 Foliated Schwarz symmetry......Page 199
8.3.1 Turning point......Page 201
8.3.2 Mountain-pass solutions......Page 202
8.3.3 Uniqueness for small λ......Page 203
8.4 The parabolic equation......Page 206
8.5.1 The p-Laplacian......Page 209
8.5.2 The biharmonic operator......Page 210
8.5.3 The fractional Laplacian......Page 211
8.5.5 Stable solutions on manifolds......Page 214
A.1 Elementary properties of the Laplace operator......Page 218
A.2 The maximum principle......Page 223
A.3 Harnack\'s inequality......Page 224
A.4 The boundary-point lemma......Page 225
A.5 Elliptic operators......Page 229
A.6 The Laplace operator with a potential......Page 231
A.7 Thin domains and unbounded domains......Page 235
A.8 Nonlinear comparison principle......Page 236
A.9.1 Linear theory and weak comparison principle......Page 237
A.9.2 The boundary-point lemma......Page 240
A.9.3 Sub- and supersolutions in the L1 setting......Page 241
B.1.1 Interior regularity......Page 248
B.1.2 Solving the Dirichlet problem on the unit ball......Page 250
B.1.3 Solving the Dirichlet problem on smooth domains......Page 252
B.2.1 Poisson\'s equation on the unit ball......Page 255
B.2.2 A priori estimates for C2,α solutions......Page 262
B.2.3 Existence of C2,α solutions......Page 264
B.3 Calderon-Zygmund estimates......Page 267
B.4 Moser iteration......Page 268
B.5 The inverse-square potential......Page 272
B.5.1 The kernel of L = –Δ – c/⎜x⎜2......Page 273
B.5.2 Functional setting......Page 274
B.5.3 The case ξ = 0......Page 275
B.5.4 The case ξ ≠ 0......Page 283
C.1.1 The isoperimetric inequality......Page 288
C.1.2 The Sobolev inequality......Page 290
C.1.3 The Hardy inequality......Page 291
C.2 Submanifolds of RN......Page 293
C.2.1 Metric tensor, tangential gradient......Page 294
C.2.2 Surface area of a submanifold......Page 296
C.2.3 Curvature, Laplace-Beltrami operator......Page 297
C.2.4 The Sobolev inequality on submanifolds......Page 302
C.3 Geometry of level sets......Page 309
C.3.1 Coarea formula......Page 310
C.4 Spectral theory of the Laplace operator on the sphere......Page 312
References......Page 318