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E-raamat: Stable Solutions of Elliptic Partial Differential Equations

Louis Dupaigne

Formaat: 335 pages
Ilmumisaeg: 15-Mar-2011
Kirjastus: Chapman & Hall/CRC
Keel: eng
ISBN-13: 9781040211007

Teised raamatud teemal:

Differential calculus & equations

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Formaat: 335 pages
Ilmumisaeg: 15-Mar-2011
Kirjastus: Chapman & Hall/CRC
Keel: eng
ISBN-13: 9781040211007

Teised raamatud teemal:

Differential calculus & equations

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Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces).

Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.

Arvustused

the author has put every effort into giving complete details of the computations. ... the self-contained one hundred-page long appendices provide a splendid resource for the reader. The material contained there has been chosen with particular good taste and constitutes in itself a valuable source of information. Remarks and further features ... are presented in an 'exercise' format. I have found many of them suitably posed and reasonable to work at the level of the book. -Jose C. Sabina de Lis, Mathematical Reviews, 2012i

Preface

1 Defining stability

(28)

1.1 Stability and the variations of energy

(8)

1.1.1 Potential wells

(4)

1.1.2 Examples of stable solutions

(4)

1.2 Linearized stability

(6)

1.2.1 Principal eigenvalue of the linearized operator

(2)

1.2.2 New examples of stable solutions

(4)

1.3 Elementary properties of stable solutions

(5)

1.3.1 Uniqueness

(1)

1.3.2 Nonuniqueness

(2)

1.3.3 Symmetry

(2)

1.4 Dynamical stability

(4)

1.5 Stability outside a compact set

(2)

1.6 Resolving an ambiguity

(3)

2 The Gelfand problem

(18)

2.1 Motivation

(1)

2.2 Dimension N = 1

(4)

2.3 Dimension N = 2

(1)

2.4 Dimension N ≥ 3

(9)

2.4.1 Stability analysis

(5)

2.5 Summary

(3)

3 Extremal solutions

(28)

3.1 Weak solutions

(3)

3.1.1 Defining weak solutions

(3)

3.2 Stable weak solutions

(7)

3.2.1 Uniqueness of stable weak solutions

(2)

3.2.2 Approximation of stable weak solutions

(5)

3.3 The stable branch

(17)

3.3.1 When is λ finite?

(1)

3.3.2 What happens at λ = λ?

(7)

3.3.3 Is the stable branch a (smooth) curve?

(4)

3.3.4 Is the extremal solution bounded?

(2)

4 Regularity theory of stable solutions

(24)

4.1 The radial case

(5)

4.2 Back to the Gelfand problem

(2)

4.3 Dimensions N = 1, 2, 3

(3)

4.4 A geometric Poincare formula

(3)

4.5 Dimension N = 4

(8)

4.5.1 Interior estimates

(6)

4.5.2 Boundary estimates

(1)

4.5.3 Proof of Theorem 4.5.1 and Corollary 4.5.1

(1)

4.6 Regularity of solutions of bounded Morse index

(3)

5 Singular stable solutions

(38)

5.1 The Gelfand problem in the perturbed ball

(11)

5.2 Flat domains

110

(5)

5.3 Partial regularity of stable solutions in higher dimensions

115

(22)

5.3.1 Approximation of singular stable solutions

116

(3)

5.3.2 Elliptic regularity in Morrey spaces

119

(4)

5.3.3 Measuring singular sets

123

(2)

5.3.4 A monotonicity formula

125

(5)

5.3.5 Proof of Theorem 5.3.1

130

(7)

6 Liouville theorems for stable solutions

137

(26)

6.1 Classifying radial stable entire solutions

137

(4)

6.2 Classifying stable entire solutions

141

(6)

6.2.1 The Liouville equation

141

(2)

6.2.2 Dimension N = 2

143

(2)

6.2.3 Dimensions N = 3, 4

145

(2)

6.3 Classifying solutions that are stable outside a compact set

147

(16)

6.3.1 The critical case

147

(7)

6.3.2 The supercritical range

154

(4)

6.3.3 Flat nonlinearities

158

(5)

7 A conjecture of De Giorgi

163

(16)

7.1 Statement of the conjecture

163

(1)

7.2 Motivation for the conjecture

164

(9)

7.2.1 Phase transition phenomena

164

(2)

7.2.2 Monotone solutions and global minimizers

166

(6)

7.2.3 From Bernstein to De Giorgi

172

(1)

7.3 Dimension N = 2

173

(1)

7.4 Dimension N = 3

174

(5)

8 Further readings

179

(24)

8.1 Stability versus geometry of the domain

179

(5)

8.1.1 The half-space

179

(2)

8.1.2 Domains with controlled volume growth

181

(2)

8.1.3 Exterior domains

183

(1)

8.2 Symmetry of stable solutions

184

(2)

8.2.1 Foliated Schwarz symmetry

184

(2)

8.2.2 Convex domains

186

(1)

8.3 Beyond the stable branch

186

(5)

8.3.1 Turning point

186

(1)

8.3.2 Mountain-pass solutions

187

(1)

8.3.3 Uniqueness for small λ

188

(3)

8.3.4 Regularity of solutions of bounded Morse index

191

(1)

8.4 The parabolic equation

191

(3)

8.5 Other energy functionals

194

(9)

8.5.1 The p-Laplacian

194

(1)

8.5.2 The biharmonic operator

195

(1)

8.5.3 The fractional Laplacian

196

(3)

8.5.4 The area functional

199

(1)

8.5.5 Stable solutions on manifolds

199

(4)

A Maximum principles

203

(30)

A.1 Elementary properties of the Laplace operator

203

(5)

A.2 The maximum principle

208

(1)

A.3 Harnack's inequality

209

(1)

A.4 The boundary-point lemma

210

(4)

A.5 Elliptic operators

214

(2)

A.6 The Laplace operator with a potential

216

(4)

A.7 Thin domains and unbounded domains

220

(1)

A.8 Nonlinear comparison principle

221

(1)

A.9 L1 theory for the Laplace operator

222

(11)

A.9.1 Linear theory and weak comparison principle

222

(3)

A.9.2 The boundary-point lemma

225

(1)

A.9.3 Sub- and supersolutions in the L1 setting

226

(7)

B Regularity theory for elliptic operators

233

(40)

B.1 Harmonic functions

233

(7)

B.1.1 Interior regularity

233

(2)

B.1.2 Solving the Dirichlet problem on the unit ball

235

(2)

B.1.3 Solving the Dirichlet problem on smooth domains

237

(3)

B.2 Schauder estimates

240

(12)

B.2.1 Poisson's equation on the unit ball

240

(7)

B.2.2 A priori estimates for C2,α solutions

247

(2)

B.2.3 Existence of C2,α solutions

249

(3)

B.3 Calderon-Zygmund estimates

252

(1)

B.4 Moser iteration

253

(4)

B.5 The inverse-square potential

257

(16)

B.5.1 The kernel of L = - Δ - c/|x|2

258

(1)

B.5.2 Functional setting

259

(1)

B.5.3 The case ξ = 0

260

(8)

B.5.4 The case ξ ≠ 0

268

(5)

C Geometric tools

273

(30)

C.1 Functional inequalities

273

(5)

C.1.1 The isoperimetric inequality

273

(2)

C.1.2 The Sobolev inequality

275

(1)

C.1.3 The Hardy inequality

276

(2)

C.2 Submanifolds of RN

278

(16)

C.2.1 Metric tensor, tangential gradient

279

(2)

C.2.2 Surface area of a submanifold

281

(1)

C.2.3 Curvature, Laplace-Beltrami operator

282

(5)

C.2.4 The Sobolev inequality on submanifolds

287

(7)

C.3 Geometry of level sets

294

(3)

C.3.1 Coarea formula

295

(2)

C.4 Spectral theory of the Laplace operator on the sphere

297

(6)

References

303

(16)

Index

319

Louis Dupaigne is an assistant professor at Université Picardie Jules Verne in Amiens, France.

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Differential equations, Elliptic - Numerical solutions

E-raamat: Stable Solutions of Elliptic Partial Differential Equations

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