Nonlinear Differential Equations of Monotone Types in Banach Spaces (eBook)

Contents 5
Preface 7
Acronyms 9
Chapter 1 Fundamental Functional Analysis 11
1.1 Geometry of Banach Spaces 11
1.2 Convex Functions and Subdifferentials 14
1.3 Sobolev Spaces and Linear Elliptic Boundary Value Problems 20
1.4 Infinite-Dimensional Sobolev Spaces 31
References 36
Chapter 2 Maximal Monotone Operators in Banach Spaces 37
2.1 Minty–Browder Theory of Maximal Monotone Operators 37
2.2 Maximal Monotone Subpotential Operators 57
2.3 Elliptic Variational Inequalities 82
2.4 Nonlinear Elliptic Problems of Divergence Type 91
Bibliographical Remarks 105
References 105
Chapter 3 Accretive Nonlinear Operators in Banach Spaces 107
3.1 Definition and General Theory 107
3.2 Nonlinear Elliptic Boundary Value Problem in Lp 116
3.3 Quasilinear Partial Differential Operators of First Order 129
Bibliographical Remarks 136
References 136
Chapter 4 The Cauchy Problem in Banach Spaces 137
4.1 The Basic Existence Results 137
4.2 Approximation and Structural Stability of Nonlinear Evolutions 178
4.3 Time-Dependent Cauchy Problems 187
4.4 Time-Dependent Cauchy Problem Versus Stochastic Equations 193
Bibliographical Remarks 199
References 200
Chapter 5 Existence Theory of Nonlinear DissipativeDynamics 203
5.1 Semilinear Parabolic Equations 203
5.2 Parabolic Variational Inequalities 219
5.3 The Porous Media Diffusion Equation 236
5.4 The Phase Field System 245
5.5 The Equation of Conservation Laws 248
5.6 SemilinearWave Equations 251
5.7 Navier–Stokes Equations 260
Bibliographical Remarks 278
References 278
Index 281