Title Page
Contents
국문초록 8
Abstract 9
Chapter 1. Introduction 11
1.1. Spectral theory of the NP operator 11
1.2. Field concentration 13
Chapter 2. Theoretical backgrouds 17
2.1. Layer potentials and the NP operator for the Laplace equation 17
2.2. Layer potentials and the NP operator for the Lamé system 20
Chapter 3. Spectrum of the NP operator and optimal estimates for transmission problems in presence of two circular inclusions 24
3.1. The NP operator for doubly connected domains and well-posedness of the problem 27
3.1.1. The NP operator for doubly connected domains 29
3.1.2. ½ eigenvectors 32
3.1.3. Well-posedness and continuous dependency 34
3.2. NP spectrum on two circles 37
3.2.1. NP spectrum on concentric circles 38
3.2.2. Möbius invariance and NP spectrum on two circles 38
3.3. Spectral representations of the solution 44
3.3.1. Vanishing integral condition 44
3.3.2. General case 50
3.4. Proofs of theorems 52
3.5. Optimality 60
Chapter 4. A concavity condition for existence of a negative value in NP spectrum in three dimensions 65
4.1. Statements of main results 66
4.2. Inversion in a sphere 66
Chapter 5. Spectral structure of the Neumann-Poincaré operator on tori 71
5.1. The NP operator in toroidal coordinate system 72
5.2. Decomposition of the NP opertor 74
5.3. Numerical range of Ak and the proof of main theorem[이미지참조] 78
Chapter 6. Spectral properties of the NP operator on rotationally symmetric domains in two dimensions 84
6.1. The NP operator and Grunsky coefficients, lemniscates 86
6.1.1. Double layer potential and Cauchy transform 86
6.1.2. The NP operator and Grunsky coefficients 87
6.1.3. Lemniscates 89
6.2. Proof of the main theorem 90
6.3. Matrix representations 94
6.4. Examples 95
6.4.1. m-star-shaped domains 95
6.4.2. Cassini oval 96
Chapter 7. Spectral properties of the NP operator and cloaking by anomalous localized resonance for the elasto-static system 99
7.1. Spectrum of elastic NP operators on smooth planar domains 99
7.1.1. Compactness of K² - k02I and spectrum[이미지참조] 100
7.1.2. Spectrum of the NP operator on disks and ellipses 102
7.2. Anomalous localized resonance and cloaking 106
7.2.1. Resonance estimates 106
7.2.2. Anomalous localized resonance on ellipses 109
7.3. Derivation of spectrum on disks and ellipses 111
7.3.1. Disks 112
7.3.2. Ellipses 114
7.4. Proofs of CALR 120
Bibliography 130