Singularities in boundary value problems

Title Page......Page 2 Copyright......Page 3 Contents......Page 4 Preface......Page 6 Introduction......Page 8 Notation......Page 12 1.1 Geometry of the domains under consideration......Page 14 1.2 Basic definitions and properties of Sobolev spaces......Page 15 1.3 Continuation from polygonal domains......Page 21 1.4 Trace theorems on polygonal domains......Page 24 1.5 Green formula on polygonal domains......Page 37 1.6 Density with boundary conditions......Page 41 1.7 Comments and remarks......Page 47 2.1 Variational solutions and classical regularity results......Page 49 2.2 Basic a-priori inequalities in polygons......Page 56 2.3 Fredholm property in 2d......Page 58 2.4 Derivation of the singular solutions in 2d......Page 69 2.5 Edge behavior in 3d ......Page 73 2.6 Vertex behavior in 3d......Page 85 2.7 Comments and remarks......Page 95 3.1 Variational solutions and classical regularity results......Page 97 3.2 Local study via Fourier transform near a corner, the Kondratiev method......Page 100 3.3 Analysis of some transcendental equations......Page 112 3.4 Derivation of the singular solutions......Page 120 3.5 Comments and remarks......Page 123 4.1 Variational solutions and classical regularity results......Page 125 4.2 Derivation of the singular solutions in 2d......Page 127 4.3 Resolvent of the Lame system......Page 132 4.4 Edge behavior in 3d......Page 147 4.5 Vertex behavior in 3d......Page 153 4.6 Comments and remarks......Page 161 5.1 Existence and uniqueness......Page 164 5.2 Corner behavior of the heat equation in 2d......Page 169 5.3 Corner behavior of the wave equation in 2d......Page 173 5.4 Comments and remarks......Page 176 6.1 Exact controllability of the wave equation......Page 177 6.2 The Stokes system as a limit case......Page 185 6.3 Some numerical aspects......Page 190 6.4 Crack propagation in brittle fracture mechanics......Page 195 6.5 Remarks on nonlinear problems......Page 201 Bibliography......Page 204 Appendix......Page 208 Notation index......Page 210 Studies the solutions of a boundary problem near corner edges and vertices. The text focuses on real-life problems considered in the actual geometry met in the applications and highlights the singular solutions which carry the main physical information.