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XVI Table of Contents

3.3 Local Coordinates ...................................... 108
3.4 Short Excursion to Elementary Differential Geometry . . . . . . . 111
3.4.1 Second Order Differential Operators
in Divergence Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.5 Distributional Derivatives and Abstract Green’s
Second Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.6 The Green Representation Formula . . . . . . . . . . . . . . . . . . . . . . . 130
3.7 Green’s Representation Formulae in Local Coordinates . . . . . . 135
3.8 Multilayer Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.9 Direct Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . 145
3.9.1 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.9.2 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4. Sobolev Spaces ........................................... 159
s
4.1 The Spaces H (Ω)...................................... 159
s
4.2 The Trace Spaces H (Γ) ................................ 169
4.2.1 Trace Spaces for Periodic Functions on a Smooth
2
Curve in IR ..................................... 181
2
4.2.2 Trace Spaces on Curved Polygons in IR ............. 185
4.3 The Trace Spaces on an Open Surface ..................... 189
4.4 Weighted Sobolev Spaces ................................ 191
5. Variational Formulations.................................. 195
5.1 Partial Differential Equations of Second Order . . . . . . . . . . . . . 195
5.1.1 Interior Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.1.2 Exterior Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.1.3 Transmission Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.2 Abstract Existence Theorems for Variational Problems . . . . . . 218
5.2.1 The Lax–Milgram Theorem . . . . . . . . . . . . . . . . . . . . . . . . 219
5.3 The Fredholm–Nikolski Theorems . . . . . . . . . . . . . . . . . . . . . . . . 226
5.3.1 Fredholm’s Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5.3.2 The Riesz–Schauder and the Nikolski Theorems . . . . . . 235
5.3.3 Fredholm’s Alternative for Sesquilinear Forms . . . . . . . . 240
5.3.4 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.4 G˚arding’s Inequality for Boundary Value Problems . . . . . . . . . 243
5.4.1 G˚arding’s Inequality for Second Order Strongly
Elliptic Equations in Ω ........................... 243
5.4.2 The Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.4.3 G˚arding’s Inequality for Exterior Second Order
Problems ........................................ 254
5.4.4 G˚arding’s Inequality for Second Order Transmission
Problems ........................................ 259
5.5 Existence of Solutions to Boundary Value Problems . . . . . . . . . 259
5.5.1 Interior Boundary Value Problems . . . . . . . . . . . . . . . . . . 260