Preface ........................................................ xi
1 Defining stability ........................................... 1
1.1 Stability and the variations of energy .................. 1
1.1.1 Potential wells .................................. 1
1.1.2 Examples of stable solutions ..................... 5
1.2 Linearized stability .................................... 9
1.2.1 Principal eigenvalue of the linearized
operator ......................................... 9
1.2.2 New examples of stable solutions ................ 11
1.3 Elementary properties of stable solutions .............. 15
1.3.1 Uniqueness ...................................... 15
1.3.2 Nonuniqueness ................................... 16
1.3.3 Symmetry ........................................ 18
1.4 Dynamical stability .................................... 20
1.5 Stability outside a compact set ........................ 24
1.6 Resolving an ambiguity ................................. 26
2 The Gelfand problem ......................................... 29
2.1 Motivation ............................................. 29
2.2 Dimension N = 1 ........................................ 30
2.3 Dimension N = 2 ........................................ 34
2.4 Dimension N ≥ 3 ........................................ 35
2.4.1 Stability analysis .............................. 39
2.5 Summary ................................................ 44
3 Extremal solutions .......................................... 47
3.1 Weak solutions ......................................... 48
3.1.1 Defining weak solutions ......................... 48
3.2 Stable weak solutions .................................. 51
3.2.1 Uniqueness of stable weak solutions ............. 51
3.2.2 Approximation of stable weak solutions .......... 53
3.3 The stable branch ...................................... 58
3.3.1 When is λ* Finite? .............................. 61
3.3.2 What happens at λ = λ*? ......................... 62
3.3.3 Is the stable branch a (smooth) curve? .......... 69
3.3.4 Is the extremal solution bounded? ............... 73
4 Regularity theory of stable solutions ....................... 75
4.1 The radial case ........................................ 75
4.2 Back to the Gelfand problem ............................ 80
4.3 Dimensions N = 1, 2, 3 ................................. 82
4.4 A geometric Poincare formula ........................... 85
4.5 Dimension N = 4 ........................................ 88
4.5.1 Interior estimates .............................. 88
4.5.2 Boundary estimates .............................. 94
4.5.3 Proof of Theorem 4.5.1 and Corollary 4.5.1 ...... 95
4.6 Regularity of solutions of bounded Morse index ......... 96
5 Singular stable solutions ................................... 99
5.1 The Gelfand problem in the perturbed ball .............. 99
5.2 Flat domains .......................................... 110
5.3 Partial regularity of stable solutions in higher
dimensions ............................................ 115
5.3.1 Approximation of singular stable solutions ..... 116
5.3.2 Elliptic regularity in Morrey spaces ........... 119
5.3.3 Measuring singular sets ........................ 123
5.3.4 A monotonicity formula ......................... 125
6 Liouville theorems for stable solutions .................... 137
6.1 Classifying radial stable entire solutions ............ 137
6.2 Classifying stable entire solutions ................... 141
6.2.1 The Liouville equation ......................... 141
6.2.2 Dimension N = 2 ................................ 143
6.2.3 Dimensions N = 3, 4 ............................ 145
6.3 Classifying solutions that are stable outside
a compact set ......................................... 147
6.3.1 The critical case .............................. 147
6.3.2 The supercritical range ........................ 154
6.3.3 Flat nonlinearities ............................ 158
7 A conjecture of De Giorgi .................................. 163
7.1 Statement of the conjecture ........................... 163
7.2 Motivation for the conjecture ......................... 164
7.2.1 Phase transition phenomena ..................... 164
7.2.2 Monotone solutions and global minimizers ....... 166
7.2.3 From Bernstein to De Giorgi .................... 172
7.3 Dimension N = 2 ....................................... 173
7.4 Dimension N = 3 ....................................... 174
8 Further readings ........................................... 179
8.1 Stability versus geometry of the domain ............... 179
8.1.1 The half-space ................................. 179
8.1.2 Domains with controlled volume growth .......... 181
8.1.3 Exterior domains ............................... 183
8.2 Symmetry of stable solutions .......................... 184
8.2.1 Foliated Schwarz symmetry ...................... 184
8.2.2 Convex domains ................................. 186
8.3 Beyond the stable branch .............................. 186
8.3.1 Turning point .................................. 186
8.3.2 Mountain-pass solutions ........................ 187
8.3.3 Uniqueness for small λ ......................... 188
8.3.4 Regularity of solutions of bounded Morse
index .......................................... 191
8.4 The parabolic equation ................................ 191
8.5 Other energy functionals .............................. 194
8.5.1 The p-Laplacian ................................ 194
8.5.2 The biharmonic operator ........................ 195
8.5.3 The fractional Laplacian ....................... 196
8.5.4 The area functional ............................ 199
8.5.5 Stable solutions on manifolds .................. 199
A Maximum principles ......................................... 203
A.1 Elementary properties of the Laplace operator ......... 203
A.2 The maximum principle ................................. 208
A.3 Harnack's inequality .................................. 209
A.4 The boundary-point lemma .............................. 210
A.5 Elliptic operators .................................... 214
A.6 The Laplace operator with a potential ................. 216
A.7 Thin domains and unbounded domains .................... 220
A.8 Nonlinear comparison principle ........................ 221
A.9 L1 theory for the Laplace operator .................... 222
A.9.1 Linear theory and weak comparison principle .... 222
A.9.2 The boundary-point lemma ....................... 225
A.9.3 Sub- and supersolutions in the L1 setting ...... 226
В Regularity theory for elliptic operators ................... 233
B.l Harmonic functions .................................... 233
B.l.l Interior regularity ............................ 233
B.l.2 Solving the Dirichlet problem on the unit
ball ........................................... 235
B.1.3 Solving the Dirichlet problem on smooth
domains ........................................ 237
B.2 Schauder estimates .................................... 240
B.2.1 Poisson's equation on the unit ball ............ 240
B.2.2 A priori estimates for C2,α solutions .......... 247
B.2.3 Existence of C2,α solutions .................... 249
B.3 Calderon-Zygmund estimates ............................ 252
B.4 Moser iteration ....................................... 253
B.5 The inverse-square potential .......................... 257
B.5.1 The kernel of L = -Δ - c/|x|2 .................. 258
B.5.2 Functional setting ............................. 259
B.5.3 The case ξ = 0 ................................. 260
B.5.4 The case ξ ≠ 0 ................................. 268
С Geometric tools ............................................ 273
C.l Functional inequalities ................................ 273
C.l.l The isoperimetric inequality ................... 273
C.l.2 The Sobolev inequality ......................... 275
C.1.3 The Hardy inequality ........................... 276
C.2 Submanifolds of  N .................................... 278
C.2.1 Metric tensor, tangential gradient ............. 279
C.2.2 Surface area of a submanifold .................. 281
C.2.3 Curvature, Laplace-Beltrami operator ........... 282
C.2.4 The Sobolev inequality on submanifolds ......... 287
C.3 Geometry of level sets ................................ 294
C.3.1 Coarea formula ................................. 295
C.4 Spectral theory of the Laplace operator on the
sphere ................................................ 297
References ................................................. 303
Index ......................................................... 319
|
