Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains (eBook)

Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains 3
Contents 6
Preface 9
Introduction 10
Chapter 1 Preliminaries 14
1.1 List of symbols 14
1.2 Operators and formulae related to spherical coordinates 15
1.3 The quasi-distance function re and its properties 17
1.4 Function spaces 18
1.5 Some inequalities 19
1.6 Sobolev embedding theorems 20
1.7 The Cauchy problem for a differential inequality 22
1.8 Additional auxiliary results 22
1.8.1 Stampacchia’s Lemma 22
1.8.2 Other assertions 23
Chapter 2 Eigenvalue problem and integro-differential inequalities 25
2.1 Eigenvalue problem for the m-Laplacian in a bounded domain on the unit sphere 25
2.2 The Friedrichs-Wirtinger type inequality 29
2.3 The Hardy and Hardy-Friedrichs-Wirtinger type inequalities 33
2.4 Auxiliary integro-differential inequalities 36
Chapter 3 Best possible estimates of solutions to the transmission problem for linear elliptic divergence second-order equations in a conical domain 41
3.1 Introduction 41
3.2 Local estimate at the boundary 45
3.3 Global integral estimates 51
3.4 Local integral weighted estimates 58
3.5 The power modulus of continuity at the conical point for weak solutions 64
3.5.1 Proof of Theorem 3.3 64
3.5.2 Remark to Theorem 3.7 66
3.6 Appendix 67
3.7 Examples 72
Chapter 4 Transmission problem for the Laplace operator with N different media 75
4.1 Introduction 75
4.2 Auxiliary statements and inequalities 81
4.2.1 The eigenvalue problem 81
4.2.2 The comparison principle 82
4.3 The barrier function. The preliminary estimate of the solution modulus 83
4.4 Local estimate at the boundary 90
4.5 Global integral estimates 90
4.6 Local integral weighted estimates 98
4.7 The power modulus of continuity at the conical point for weak solutions 102
4.8 Appendix: Eigenvalue transmission problem in a composite plane domain with an angular point 104
4.8.1 Four-media transmission problem 106
4.8.2 Three-media transmission problem 109
4.8.3 Two-media transmission problem 111
Chapter 5 Transmission problem for weak quasi-linear elliptic equations in a conical domain 112
5.1 Introduction 112
5.2 Local estimate at the boundary 115
5.3 Global integral estimate 115
5.4 Local integral weighted estimates 121
5.5 The power modulus of continuity at the conical point for weak solutions 127
5.5.1 Proof of Theorem 5.3. 127
5.5.2 Remark to Theorem 5.5 129
5.6 Example 130
Chapter 6 Transmission problem for strong quasi-linear elliptic equations in a conical domain 141
6.1 Introduction 141
6.2 Comparison principle 145
6.3 Maximum principle 147
6.4 Local estimate at the boundary 154
6.5 Integral estimates 161
6.6 The power modulus of continuity at the conical point for weak solutions 167
6.7 Appendix 168
6.7.1 The barrier function. The preliminary estimate of the solution modulus 168
Chapter 7 Best possible estimates of solutions to the transmission problem for aquasi-linear elliptic divergence second-order equation in a domain with a boundary edge 176
7.1 Introduction. Assumptions 176
7.2 The comparison principle 181
7.3 Construction of the barrier function 182
7.4 The case .+/a+= .-/a- 185
7.4.1 The barrier function 185
7.4.2 Properties of the eigenvalue . for (CPE) 190
7.4.3 Perturbation of problem (MiP) 192
7.4.4 Estimates of the (TDQL) solution modulus 195
7.4.5 Example 202
7.5 The case .+/a+ = .-/a- 203
7.5.1 Properties of solutions to the Sturm-Liouville boundary problem (StL) 203
7.5.2 Estimates of the (TDQL) solution modulus 208
Bibliography 213
Index 219
Notation Index 221