CONTEMPORARY MATHEMATICS 362 Partial Differential Equations and Inverse Problems Pan-American Advanced Studies Institute on Partial Differential Equations, Nonlinear Analysis and Inverse Problems January 6-18, 2003 Santiago, Chile Carlos Conca Raul Man6sevich Gunther Uhlmann Michael S. Vogelius Editors http://dx.doi.org/10.1090/conm/362 Partial Differential Equations and Inverse Problems CoNTEMPORARY MATHEMATICS 362 Partial Differential Equations and Inverse Problems Pan-American Advanced Studies Institute on Partial Differential Equations, Nonlinear Analysis and Inverse Problems January 6-18,2003 Santiago, Chile Carlos Conca Raul Man6sevich Gunther Uhlmann Michael S. Vogelius Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius This volume represents the Proceedings of the Pan-American Advanced Studies Insti- tute on Partial Differential Equations, Nonlinear Analysis and Inverse Problems held in Santiago, Chile, January 6-18, 2003, with support from the U.S. National Science Foun- dation, the U.S. Department of Energy, the Chilean Conicyt, Centro de mode lamiento Matematico (CMM), and the Canadian Pacific Institute of Mathematical Sciences (PIMS). 2000 Mathematics Subject Classification. Primary 05C40, 35B60, 35B65, 35P25, 35P99, 35Q40, 35Q53, 35Q55, 35Q60, 35R30, 58J05, 78A48, 81Q10, 81U40. Library of Congress Cataloging-in-Publication Data Pan-American Advanced Studies Institute on Partial Differential Equations, Nonlinear Analysis and Inverse Problems (2003 : Santiago, Chile) Partial differential equations and inverse problems : Pan-American Advanced Studies Institute on Partial Differential Equations, Nonlinear Analysis and Inverse Problems, January 6-18, 2003, Santiago, Chile / Carlos Conca ... [et al.], editors. p. em. Includes bibliographical references. ISBN 0-8218-3448-7 (alk. paper) 1. Differential equations, Partial-Congresses. 2. Inverse problems (Differential equations)- Congresses. I. Conca, Carlos, II. Title. QA374.P26 2003 5151.353-dc22 2004051901 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. 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Contact the AMS for copyright status of individual articles. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04 Contents Preface vii List of Participants ix Inverse Scattering on the Line with Incomplete Scattering Data TUNCAY AKTOSUN 1 On Universal Inequalities for the Low Eingenvalues of the Buckling Problem MARKS. ASHBAUGH 13 Continuous and Discrete Inverse Conductivity Problems JOHN BARAS, CARLOS BERENSTEIN, and FRANKLIN GAVILANEZ 33 Connection between the Lieb-Thirring Conjecture for Schrodinger Operators and an Isoperimetric Problem for Ovals on the Plane RAFAEL D. BENGURIA and MICHAEL Loss 53 An Introduction to PDE Methods in Finance JEROME BuscA 63 A Review of Some Recent Work on Impedance Imaging for Inhomogeneities of Low Volume Fraction YVES CAPDEBOSCQ and MICHAEL S. VOGELIUS 69 Matrix Bispectrality and Huygens' Principle for Dirac Operators FABIO A. C. C. CHALUB and JORGE P. ZUBELLI 89 Well-Posedness for the Schrodinger-Debye Equation A. J. CORCHO and F. LINARES 113 On Domains for which the Clamped Plate System is Positivity Preserving A. DALL' AcQUA and G. SwEERS 133 Entropy Dissipation and Wasserstein Metric Methods for the Viscous Burgers' Equation: Convergence to Diffusive Waves MARCO DI FRANCESCO and PETER A. MARKOWICH 145 Remarks about the Flashing Rachet JEAN DOLBEAULT, DAVID KINDERLEHRER, and MICHAL KOWALCZYK 167 A Least Action Principle for Steepest Descent in a Non-Convex Landscape NASSIF GHOUSSOUB and ROBERT J. McCANN 177 v vi CONTENTS A Network Tomography Problem Related to the Hypercube F. ALBERTO GRUNBAUM and LAURA FELICIA MATUSEVICH 189 On the Structure of the Schrodinger Propagator ANDREW HASSELL and JARED WUNSCH 199 The Downward Continuation Approach to Modeling and Inverse Scattering of Seismic Data in the Kirchhoff Approximation. MAARTEN V. DE HOOP 211 Approximate Invariant Manifold of the Allen-Cahn Flow in Two Dimensions MICHAL KOWALCZYK 233 On Some Spectral Problems of Mathematical Physics PETER KUCHMENT 241 Inverse Problem for a Random Potential MATTI LASSAS, LASSI PAIVARINTA, and EERO SAKSMAN 277 Pseudodifferential Analysis for the Laplacian on Noncompact Symmetric Spaces. RAFE MAZZEO 289 Boundary Structure and Cohomology of b-Complex Manifolds GERARDO A. MENDOZA 303 Unique Continuation Property for Elliptic Systems and Crack Determination in Anisotropic Elasticity GEN NAKAMURA, GUNTHER UHLMANN, and JENN-NAN WANG 321 Duality in Sub-Supercritical Bubbling in the Brezis-Nirenberg Problem near the Critical Exponent. MANUEL DELPINO, JEAN DOLBEAULT, and MONICA Musso 339 High-Order Domain Variations in Boundary Value and Free Boundary Problems FERNANDO REITICH 351 Radiation Fields and Inverse Scattering on Asymptotically Euclidean Manifolds ANTONIO SA BARRETO 371 Interactive Oscillation Sources in Signorini's Type Problems CLAUDIA TIMOFTE and CARLOS CONCA 381 Inverse Scattering with Time-Periodic Potentials RICARDO WEDER 393 A Local Borg-Marchenko Theorem for Difference Equations with Complex Coefficients. Rum WEIKARD 403 Preface This volume represents the Proceedings of the Pan-American Advanced Studies Institute on Partial Differential Equations, Nonlinear Analysis and Inverse Prob- lems, held in Santiago, Chile, January 6-18, 2003. The Pan-American Advanced Studies Institutes (PASis) are programs initiated and supported by the US National Science Foundation and the US Department of Energy. The objective of a PASI is "to disseminate advanced scientific and engineering knowledge and stimulate train- ing and cooperation among researchers of the Americas in the physical sciences and engineering fields". The PASis are modeled on the NATO Advanced Studies Insti- tutes. This particular PASI, whose proceedings make up this volume, was held at, and also partially sponsored by the Centro de Modelamiento Matematico ( CMM), Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile. It also formed one of the workshops of the 2003 Thematic Year on Inverse Problems sponsored by the Pacific Institute of Mathematical Sciences (PIMS). The organizers were Rafael Benguria (Pontificia Universidad Cat6lica de Chile), Carlos Conca (Universidad de Chile), Nassif Ghoussoub (University of British Columbia), Raul Manasevich (Universidad de Chile), Wei-Ming Ni (University of Minnesota), Gunther Uhlmann (University of Washington) and Michael Vogelius (Rutgers). This Chilean PASI was a follow up to a smaller PASI held in November 2001, at the Mathematics Sciences Research Institute (MSRI), Berkeley, as one of the activities of the Inverse Problems and Applications Program that took place at MSRI during the fall of 2001. The interaction between Partial Differential Equations (PDE), Non-Linear Anal- ysis (NLA), and Inverse Problems (IP) has produced remarkable developments in the last couple of decades. One of the main objectives of the PASIon PDE, NLA and IP was to present many of these developments to advanced graduate students, postdocs and other scientists in the Americas interested in these fields and their applications. Another important objective of the PASI was to foster international cooperation throughout the Americas by bringing researchers with different areas of expertise in PDE, NLA and IP together at one event. More than 160 participants representing most countries of the Americas and a few European countries assured that this unique event was highly successful in regards to both of these objectives. During the first week of the PASI there were a series of introductory minicourses specifically intended for graduate students and postdocs. These minicourses were given by Jean-Bernard Baillon (Universidad de Chile), Jerome Busca (Universite Paris IX, Dauphine), Luis Caffarelli (University of Texas, Austin), Maarten de Hoop (Colorado School of Mines), F. Alberto Grunbaum (University of California, Berke- ley), Peter Kuchment (Texas A&M), Yan Yan Li (Rutgers) and Michael Vogelius (Rutgers). Included in this volume are five survey articles that each corresponds vii viii PREFACE