Preface p. xi
Defining stability p. 1
Stability and the variations of energy p. 1
Potential wells p. 1
Examples of stable solutions p. 5
Linearized stability p. 9
Principal eigenvalue of the linearized operator p. 9
New examples of stable solutions p. 11
Elementary properties of stable solutions p. 15
Uniqueness p. 15
Nonuniqueness p. 16
Symmetry p. 18
Dynamical stability p. 20
Stability outside a compact set p. 24
Resolving an ambiguity p. 26
The Gelfand problem p. 29
Motivation p. 29
Dimension N = 1 p. 30
Dimension N = 2 p. 34
Dimension N ≥ 3 p. 35
Stability analysis p. 39
Summary p. 44
Extremal solutions p. 47
Weak solutions p. 48
Defining weak solutions p. 48
Stable weak solutions p. 51
Uniqueness of stable weak solutions p. 51
Approximation of stable weak solutions p. 53
The stable branch p. 58
When is 驴* finite? p. 61
What happens at 驴 = 驴*? p. 62
Is the stable branch a (smooth) curve? p. 69
Is the extremal solution bounded? p. 73
Regularity theory of stable solutions p. 75
The radial case p. 75
Back to the Gelfand problem p. 80
Dimensions N = 1, 2, 3 p. 82
A geometric Poincar茅 formula p. 85
Dimension N = 4 p. 88
Interior estimates p. 88
Boundary estimates p. 94
Proof of Theorem 4.5.1 and Corollary 4.5.1 p. 95
Regularity of solutions of bounded Morse index p. 96
Singular stable solutions p. 99
The Gelfand problem in the perturbed ball p. 99
Flat domains p. 110
Partial regularity of stable solutions in higher dimensions p. 115
Approximation of singular stable solutions p. 116
Elliptic regularity in Morrey spaces p. 119
Measuring singular sets p. 123
A monotonicity formula p. 125
Proof of Theorem 5.3.1 p. 130
Liouville theorems for stable solutions p. 137
Classifying radial stable entire solutions p. 137
Classifying stable entire solutions p. 141
The Liouville equation p. 141
Dimension N = 2 p. 143
Dimensions N = 3, 4 p. 145
Classifying solutions that are stable outside a compact set p. 147
The critical case p. 147
The supercritical range p. 154
Flat nonlinearities p. 158
A conjecture of De Giorgi p. 163
Statement of the conjecture p. 163
Motivation for the conjecture p. 164
Phase transition phenomena p. 164
Monotone solutions and global minimizers p. 166
From Bernstein to De Giorgi p. 172
Dimension N = 2 p. 173
Dimension N = 3 p. 174
Further readings p. 179
Stability versus geometry of the domain p. 179
The half-space p. 179
Domains with controlled volume growth p. 181
Exterior domains p. 183
Symmetry of stable solutions p. 184
Foliated Schwarz symmetry p. 184
Convex domains p. 186
Beyond the stable branch p. 186
Turning point p. 186
Mountain-pass solutions p. 187
Uniqueness for small 驴 p. 188
Regularity of solutions of bounded Morse index p. 191
The parabolic equation p. 191
Other energy functionals p. 194
The p-Laplacian p. 194
The biharmonic operator p. 195
The fractional Laplacian p. 196
The area functional p. 199
Stable solutions on manifolds p. 199
Maximum principles p. 203
Elementary properties of the Laplace operator p. 203
The maximum principle p. 208
Harnack's inequality p. 209
The boundary-point lemma p. 210
Elliptic operators p. 214
The Laplace operator with a potential p. 216
Thin domains and unbounded domains p. 220
Nonlinear comparison principle p. 221
L1theory for the Laplace operator p. 222
Linear theory and weak comparison principle p. 222
The boundary-point lemma p. 225
Sub- and supersolutions in the L1setting p. 226
Regularity theory for elliptic operators p. 233
Harmonic functions p. 233
Interior regularity p. 233
Solving the Dirichlet problem on the unit ball p. 235
Solving the Dirichlet problem on smooth domains p. 237
Schauder estimates p. 240
Poisson's equation on the unit ball p. 240
A priori estimates for C2,asolutions p. 247
Existence of C2,asolutions p. 249
Calderon-Zygmund estimates p. 252
Moser iteration p. 253
The inverse-square potential p. 257
The kernel of L = -驴- $$$ p. 258
Functional setting p. 259
The case 驴 = 0 p. 260
The case 驴 = 0 p. 268
Geometric tools p. 273
Functional inequalities p. 273
The isoperimetric inequality p. 273
The Sobolev inequality p. 275
The Hardy inequality p. 276
Submanifolds of RN p. 278
Metric tensor, tangential gradient p. 279
Surface area of a submanifold p. 281
Curvature, Laplace-Beltrami operator p. 282
The Sobolev inequality on submanifolds p. 287
Geometry of level sets p. 294
Coarea formula p. 295
Spectral theory of the Laplace operator on the sphere p. 297
References p. 303
Index p. 319