Self-dual partial differential systems and their variational principles / Nassif Ghoussoub.

Format:

Book

Author/Creator:

Ghoussoub, N. (Nassif), 1953-

Series:

Springer monographs in mathematics

Language:

English

Subjects (All):

Differential equations, Partial.

Variational principles.

Physical Description:

xiv, 354 pages ; 25 cm.

Place of Publication:

New York ; London : Springer, 2008.

Summary:

Based on recent research by the author and his graduate students, Abbas Moameni, Leo Tzou, and Ramon Zarate, this text describes novel variational formulations and resolutions of a large class of partial differential equations and evolutions, many of which are not amenable to the methods of classical calculus of variations. While it contains recent research material, the general and unifying framework of the approach, its versatility in solving a disparate set of equations, and its reliance on basic functional analytic principles, makes it suitable for an intermediate level graduate course. The applications, however, require a fair knowledge of clasical analysis and PDEs which is needed to make judicious choices of function spaces where the self-dual variational principles apply. It is the author's hope that this material will become standard for all graduate students interested in convexity methods for PDEs.

Contents:

Part I Convex Analysis on Phase Space

2 Legendre-Fenchel Duality on Phase Space 25

2.1 Basic notions of convex analysis 25

2.2 Subdifferentiability of convex functions 26

2.3 Legendre duality for convex functions 28

2.4 Legendre transforms of integral functions 31

2.5 Legendre transforms on phase space 32

2.6 Legendre transforms on various path spaces 38

2.7 Primal and dual problems in convex optimization 45

3 Self-dual Lagrangians on Phase Space 49

3.1 Invariance under Legendre transforms up to an automorphism 49

3.2 The class of self-dual Lagrangians 51

3.3 Self-dual Lagrangians on path spaces 55

3.4 Uniform convexity of self-dual Lagrangians 57

3.5 Regularization of self-dual Lagrangians 59

3.6 Evolution triples and self-dual Lagrangians 62

4 Skew-Adjoint Operators and Self-dual Lagrangians 67

4.1 Unbounded skew-symmetric operators and self-dual Lagrangians 67

4.2 Green-Stokes formulas and self-dual boundary Lagrangians 73

4.3 Unitary groups associated to skew-adjoint operators and self-duality 78

5 Self-dual Vector Fields and Their Calculus 83

5.1 Vector fields derived from self-dual Lagrangians 84

5.2 Examples of B-self-dual vector fields 86

5.3 Operations on self-dual vector fields 88

5.4 Self-dual vector fields and maximal monotone operators 91

Part II Completely Self-dual Systems and Their Lagrangians

6 Variational Principles for Completely Self-dual Functions 99

6.1 The basic variational principle for completely self-dual functions 99

6.2 Complete self-duality in non-sefadjoint Dirichlet problems 103

6.3 Complete self-duality and non-potential PDEs in divergence form 107

6.4 Completely self-dual functionals for certain differential systems 110

6.5 Complete self-duality and semilinear transport equations 113

7 Semigroups of Contractions Associated to Self-dual Lagrangians 119

7.1 Initial-value problems for time-dependent Lagrangians 120

7.2 Initial-value parabolic equations with a diffusive term 125

7.3 Semigroups of contractions associated to self-dual Lagrangians 129

7.4 Variational resolution for gradient flows of semiconvex functions 135

7.5 Parabolic equations with homogeneous state-boundary conditions 137

7.6 Variational resolution for coupled flows and wave-type equations 140

7.7 Variational resolution for parabolic-ellipic variational inequalities 143

8 Iteration of Self-dual Lagrangians and Multiparameter Evolutions 147

8.1 Self-duality and nonhomogeneous boundary value problems 148

8.2 Applications to PDEs involving the transport operator 153

8.3 Initial-value problems driven by a maximal monotone operator 155

8.4 Lagrangian intersections of convex-concave Hamiltonian systems 161

8.5 Parabolic equations with evolving state-boundary conditions 162

8.6 Multiparameter evolutions 166

9 Direct Sum of Completely Self-dual Functionals 175

9.1 Self-dual systems of equations 176

9.2 Lifting self-dual Lagrangians to A[subscript H superscript 2 0, T] 178

9.3 Lagrangian intersections via self-duality 180

10 Semilinear Evolution Equations with Self-dual Boundary Conditions 187

10.1 Self-dual variational principles for parabolic equations 187

10.2 Parabolic semilinear equations without a diffusive term 191

10.3 Parabolic semilinear equation with a diffusive term 195

10.4 More on skew-adjoint operators in evolution equations 199

Part III Self-dual Systems and Their Antisymmetric Hamiltonians

11 The Class of Antisymmetric Hamiltonians 205

11.1 The Hamiltonian and co-Hamiltonians of self-dual Lagrangians 206

11.2 Regular maps and antisymmetric Hamiltonians 210

11.3 Self-dual functionals 212

12 Variational Principles for Self-dual Functionals and First Applications 217

12.1 Ky Fan's min-max principle 217

12.2 Variational resolution for general nonlinear equations 220

12.3 Variational resolution for the stationary Navier-Stokes equations 227

12.4 A variational resolution for certain nonlinear systems 230

12.5 A nonlinear evolution involving a pseudoregular operator 232

13 The Role of the Co-Hamiltonian in Self-dual Variational Problems 241

13.1 A self-dual variational principle involving the co-Hamiltonian 241

13.2 The Cauchy problem for Hamiltonian flows 242

13.3 The Cauchy problem for certain nonconvex gradient flows 247

14 Direct Sum of Self-dual Functionals and Hamiltonian Systems 253

14.1 Self-dual systems of equations 253

14.2 Periodic orbits of Hamiltonian systems 260

14.3 Lagrangian intersections 267

14.4 Semiconvex Hamiltonian systems 270

15 Superposition of Interacting Self-dual Functionals 275

15.1 The superposition in terms of the Hamiltonians 275

15.2 The superposition in terms of the co-Hamiltonians 278

15.3 The superposition of a Hamiltonian and a co-Hamilonian 281

Part IV Perturbations Of Self-dual Systems

16 Hamiltonian Systems of Partial Differential Equations 287

16.1 Regularity and compactness via self-duality 288

16.2 Hamiltonian systems of PDEs with self-dual boundary conditions 289

16.3 Nonpurely diffusive Hamiltonian systems of PDEs 299

17 The Self-dual Palais-Smale Condition for Noncoercive Functionals 305

17.1 A self-dual nonlinear variational principle without coercivity 306

17.2 Superposition of a regular map with an unbounded linear operator 309

17.3 Superposition of a nonlinear map with a kew-adjoint operator modulo boundary terms 315

18 Navier-Stokes and other Self-dual Nonlinear Evolutions 319

18.1 Elliptic perturbations of self-dual functionals 319

18.2 A self-dual variational principle for nonlinear evolutions 323

18.3 Navier-Stokes evolutions 331

18.4 Schodinger evolutions 340

18.5 Noncoercive nonlinear evolutions 342.

Notes:

Includes bibliographical references (pages 345-351) and index.

ISBN:

9780387848969

0387848967

OCLC:

258078803