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Mathematical treatment for thermoelastic plate with a curvilinear hole in S-plane

100%
El-Bary A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2006
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tom 26
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nr 1
77-86
EN
The Cauchy integral method has been applied to derive exact and closed expressions for Goursat's functions for the first and second fundamental problems for an infinite thermoelastic plate weakened by a hole having arbitrary shape. The plate considered is conformally mapped to the area of the right half-plane. Many previous discussions of various authors can be considered as special cases of this work. The shape of the hole being an ellipse, a crescent, a triangle, or a cut having the shape of a circular arc are included as special cases.
2
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On boundary behavior of Cauchy integrals

100%
Shiga H.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2013
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tom 67
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nr 1
EN
In this paper, we shall estimate the growth order of the n-th derivative Cauchy integrals at a point in terms of the distance between the point and the boundary of the domain. By using the estimate, we shall generalize Plemelj–Sokthoski theorem. We also consider the boundary behavior of generalized Cauchy integrals on compact bordered Riemann surfaces.
3
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The generalized Neumann-Poincaré operator and its spectrum

40%
Partyka D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction..........................................................................................................................................................................5 Preliminaries. Complex harmonic functions..........................................................................................................................7 I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19  1.1. On a boundary integral..............................................................................................................................................20  1.2. The generalized Cauchy singular integral operator $C_𝕍$.......................................................................................23  1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28  1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31  1.5. The generalized Neumann-Poincaré operator $N_𝕍$...............................................................................................36 II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41  2.1. The Douady-Earle extension $E_γ$..........................................................................................................................42  2.2. On an approximation of the Hersch-Pfluger distortion function $Φ_K$......................................................................46  2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48  2.4. The Hilbert space H...................................................................................................................................................54  2.5. The linear operator $B_γ$.........................................................................................................................................60 III. The generalized harmonic conjugation operator............................................................................................................64  3.1. The generalized harmonic conjugation operator $A_γ$.............................................................................................64  3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73  3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80  3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84 IV. Spectral values of a quasicircle.....................................................................................................................................90  4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91  4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93  4.3. Plemelj's formula for a quasicircle..............................................................................................................................96  4.4. The main spectral theorem for quasicircles.............................................................................................................103  4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108 Appendix. The inner completion of pseudo-normed spaces............................................................................................114 References......................................................................................................................................................................117 List of symbols.................................................................................................................................................................122 Index................................................................................................................................................................................124
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