RECONSTRUCTION OF SMALL INHOMOGENEITIES FROM BOUNDARY MEASUREMENTS

By WILLIAM THOMPSON

CONTENTS PART I DETECTION OF SMALL CONDUCTIVITY INCLUSIONS TRANSMISSION PROBLEM SOME NOTATIONS AND PRELIMINARIES LAYER POTENTIALS FOR THE LAPLACIAN NEUMANN AND DIRICHLET FUNCTIONS REPRESENTATION FORMULA ENERGY IDENTITIES GENERALIZED POLARIZATION TENSORS DE?NITION UNIQUENESS RESULT SYMMETRY AND POSITIVITY OF GPT’S BOUNDS FOR THE POLARIZATION TENSOR OF P´OLYA–SZEG¨O ESTIMATES OF THE WEIGHTED VOLUME AND THE CENTER OF MASS POLARIZATION TENSORS OF MULTIPLE INCLUSIONS PROPERTIES REPRESENTATION BY EQUIVALENT ELLIPSES DERIVATION OF THE FULL ASYMPTOTIC FORMULA ENERGY ESTIMATES ASYMPTOTIC EXPANSION CLOSELY SPACED SMALL INCLUSIONS DETECTION OF INCLUSIONS CONSTANT CURRENT PROJECTION ALGORITHM QUADRATIC ALGORITHM LEAST-SQUARES ALGORITHM VARIATIONAL ALGORITHM LINEAR SAMPLING METHOD LIPSCHITZ-CONTINUOUS DEPENDENCE AND MOMENTS ESTIMATIONS LIPSCHITZ-CONTINUOUS DEPENDENCE MOMENTS ESTIMATIONS PART II DETECTION OF SMALL ELASTIC INCLUSIONS TRANSMISSION PROBLEM FOR ELASTOSTATICS LAYER POTENTIALS FOR THE LAM´E SYSTEM KELVIN MATRIX UNDER UNITARY TRANSFORMS COMPLEX REPRESENTATION OF DISPLACEMENT VECTORS ELASTIC MOMENT TENSOR ASYMPTOTIC EXPANSION IN FREE SPACE PROPERTIES OF EMT’S EMT’S UNDER LINEAR TRANSFORMS EMT’S FOR ELLIPSES EMT’S FOR ELLIPTIC HOLES AND HARD ELLIPSES DERIVATION OF FULL ASYMPTOTIC EXPANSIONS FULL ASYMPTOTIC EXPANSIONS DETECTION OF EMT’S REPRESENTATION OF THE EMT’S BY ELLIPSES DETECTION OF THE LOCATION NUMERICAL RESULTS PART III DETECTION OF SMALL ELECTROMAGNETIC INCLUSIONS WELL-POSEDNESS 10 1 EXISTENCE AND UNIQUENESS OF A SOLUTION REPRESENTATION OF SOLUTIONS 11 1 PRELIMINARY RESULTS 11 2 REPRESENTATION FORMULAE DERIVATION OF ASYMPTOTIC FORMULAE 12 1 ASYMPTOTIC EXPANSION RECONSTRUCTION ALGORITHMS 13 1 ASYMPTOTIC EXPANSION 13 2 RECONSTRUCTION OF MULTIPLE INCLUSIONS 13 2 1 THE FOURIER TRANSFORM ALGORITHM 13 2 2 THE MUSIC ALGORITHM