## Strongly Elliptic Systems and Boundary Integral EquationsPartial differential equations provide mathematical models of many important problems in the physical sciences and engineering. This book treats one class of such equations, concentrating on methods involvingthe use of surface potentials. It provides the first detailed exposition of the mathematical theory of boundary integral equations of the first kind on non-smooth domains. Included are chapters on three specific examples: the Laplace equation, the Helmholtz equation and the equations of linear elasticity.The book is designed to provide an ideal preparation for studying the modern research literature on boundary element methods. |

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### Contents

Introduction | 1 |

Exercises | 15 |

Abstract Linear Equations | 17 |

The Kernel and Image | 18 |

Duality | 20 |

Compactness | 27 |

Fredholm Operators | 32 |

Hilbert Spaces | 38 |

The Third Green Identity | 200 |

Jump Relations and Mapping Properties | 202 |

Duality Relations | 211 |

Exercises | 215 |

Boundary Integral Equations | 217 |

Integral Representations | 219 |

The Dirichlet Problem | 226 |

The Neumann Problem | 229 |

Coercivity | 42 |

Elementary Spectral Theory | 45 |

Exercises | 52 |

Sobolev Spaces | 57 |

Convolution | 58 |

Differentiation | 61 |

Schwartz Distributions | 64 |

Fourier Transforms | 69 |

Sobolev Spaces First Definition | 73 |

Sobolev Spaces Second Definition | 75 |

Equivalence of the Norms | 79 |

Localisation and Changes of Coordinates | 83 |

Density and Imbedding Theorems | 85 |

Lipschitz Domains | 89 |

Sobolev Spaces on the Boundary | 96 |

The Trace Operator | 100 |

VectorValued Functions | 106 |

Exercises | 107 |

Strongly Elliptic Systems | 113 |

Strongly Elliptic Operators | 118 |

Boundary Value Problems | 128 |

Regularity of Solutions | 133 |

The Transmission Property | 141 |

Estimates for the SteklovPoincare Operator | 145 |

Exercises | 156 |

Homogeneous Distributions | 158 |

FinitePart Integrals | 159 |

Extension from ℝⁿ 0 to ℝⁿ | 166 |

Fourier Transforms | 169 |

Change of Variables | 174 |

FinitePart Integrals on Surfaces | 181 |

Exercises | 187 |

Surface Potentials | 191 |

Parametrices | 192 |

Fundamental Solutions | 197 |

Mixed Boundary Conditions | 231 |

Exterior Problems | 234 |

Regularity Theory | 239 |

Exercises | 241 |

The Laplace Equation | 246 |

Fundamental Solutions | 247 |

Spherical Harmonics | 250 |

Behaviour at Infinity | 258 |

Solvability for the Dirichlet Problem | 260 |

Solvability for the Neumann Problem | 266 |

Exercises | 268 |

The Helmholtz Equation | 276 |

Separation of Variables | 277 |

The Sommerfeld Radiation Condition | 280 |

Uniqueness and Existence of Solutions | 286 |

A Boundary Integral Identity | 289 |

Exercises | 293 |

Linear Elasticity | 296 |

Korns Inequality | 297 |

Fundamental Solutions | 299 |

Uniqueness Results | 301 |

Exercises | 305 |

Extension Operators for Sobolev Spaces | 309 |

Exercises | 315 |

Interpolation Spaces | 317 |

The KMethod | 318 |

The JMethod | 321 |

Interpolation of Sobolev Spaces | 329 |

Exercises | 333 |

Further Properties of Spherical Harmonics | 334 |

Exercises | 338 |

341 | |

347 | |

353 | |

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### Common terms and phrases

apply argument assume Banach space bounded Chapter choose closed coefficients coercive compact complete condition consider constant continuous converges Corollary Deduce define definition denote dense derivative differential Dirichlet distribution eigenvalue equivalent estimate Exercise exists extension fact Finally finite follows formula Fredholm function fundamental solution given gives grad Green identity Hence holds homogeneous function homogeneous of degree implies inequality inverse Lemma linear operator Lipschitz domain mapping property norm obtain operator orthogonal particular positive potential problem Proof prove Recall representation result satisfies self-adjoint sequence Show Sobolev spaces solution subset subspace sufficiently Suppose surface taking term Theorem theory unique vector write zero