Applied Functional Analysis and Partial Differential Equations

Cover......Page 1
S Title......Page 2
Applied Functional Analysis and Partial Differential Equations ......Page 4
ISBN  981-02-3535-6 ......Page 5
Contents ......Page 6
Preface ......Page 10
1.1  Metric  Spaces ......Page 12
1.2  Vector  Spaces ......Page 17
1.3  Banach  Spaces ......Page 19
1.4  Linear  Operators ......Page 22
1.5  Duals ......Page 27
1.6  Spectrum ......Page 34
1.7  Compact  Linear  Operators ......Page 40
1.8  Boundary  Value  Problems  for  Linear  ODEs ......Page 50
1.9  Exercises ......Page 54
2.1  Orthonormal  Sets ......Page 58
2.2  Adjoint s ......Page 65
2.3  Accretive  Operators ......Page 69
2.4  Weak  Solutions......Page 73
2.5  Example:  Constant  Coefficient  PDEs ......Page 79
2.6  Self-adjoint  Operators ......Page 81
2.7  Example:  Sturm-Liouville  Problem ......Page 87
2.8  Sectorial  Forms ......Page 91
2.9  Example:  Harmonic  Oscillator  and  Hermite  Func­tions......Page 99
2.10  Example:  Completeness  of  Bessel  Functions ......Page 103
2.11  Example:  Finite  Element  Method ......Page 107
2.12  Friedrichs  Extension......Page 110
2.13  Exercises ......Page 115
3.1  Introduction......Page 120
3.2  Fourier  Transform ......Page 125
3.3  Distributions ......Page 133
3.4  Weak  Derivatives ......Page 137
3.5  Definition  and  Basic  Properties  of  Sobolev  Spaces ......Page 146
3.6  Imbeddings  of  W^m,p(Omega) ......Page 151
3.7  Elliptic  Problems ......Page 164
3.8  Regularity  of  Weak  Solutions ......Page 171
3.9  Exercises ......Page 174
4.1  Introduction......Page 178
4.2  Bochner  Integral......Page 182
4.3  Basic Properties of Semigroups......Page 187
4.4  Example:  Wave  Equation ......Page 198
4.5  Sectorial  Operators  and  Analytic  Semigroups......Page 203
4.6  Invariant  Subspaces......Page 215
4.7  The  Inhomogeneous  Problem  -  Part  I......Page 220
4.8  Exercises......Page 223
5.1  Introduction......Page 226
5.2  Basic  Theory......Page 229
5.3  Example:  Nonlinear  Heat  Equation......Page 233
5.4  Approximation  for  Evolution  Equations......Page 235
5.5  Example:  Finite  Difference  Method......Page 244
5.6  Example:  Galerkin  Method  for  Parabolic  Equations......Page 247
5.7  Example:  Galerkin  Method  for  the  Wave  Equation......Page 250
5.8  Friedrichs  Extension  and  Galerkin  Approximations......Page 254
5.9  Exercises ......Page 256
6.1  Fractional  Powers  of  Operators......Page 258
6.2  The  Inhomogeneous  Problem  -  Part  II ......Page 265
6.3  Global  Version......Page 268
6.4  Main  Results......Page 270
6.5  Example:  Navier-Stokes  Equations......Page 278
6.6  Example:  A  Stability  Problem......Page 282
6.7  Example:  A  Classical  Solution......Page 284
6.8  Dynamical  Systems......Page 287
6.9  Example:  The  Chafee-Infante  Problem......Page 290
6.10  Exercises......Page 293
Bibliography ......Page 296
List  of  Symbols......Page 300
Index ......Page 302