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 , 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

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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 (), 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 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 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 to such a minicourse. During the second week the PASI focused on recent devel- opments in the interaction between IP, NLA and PDE. This week covered a very broad range of topics within the three subjects, as is also amply reflected by the many, high quality contributions to these proceedings. In closing the preface we would like to thank the sponsors (The US NSF, the US DOE, the Chilean Conicyt, CMM, and the Canadian PIMS) for their generous support of this PASI. In particular, we should like to thank Harold Stolberg of NSF for his advice and guidance during the planning and funding phases. We should also like to thank the staff of the CMM, especially Gladys Cavallone, without whose dedicated effort this could never have become such a highly successful event.

Carlos Conca Raul Manasevich Gunther Uhlmann Michael Vogelius List of Participants

Aceff, Flor Benguria, Rafael Universidad Nacional Autonoma de Pontificia Universidad Catolica de Mexico, Mexico Chile, Chile Aduen, Hugo Berenstein, Carlos Universidad Nacional de Colombia, University of Maryland, USA Medellin, Colombia Berestycki, Henri Aktosun, Tuncay Ecole des Hautes Etudes en Sciences Mississippi State University, USA Sociales, Alarcon, Salomon Bocea, Marian Universidad de Chile, Chile Carnegie Mellon University, USA Alvarez, Felipe Borgna, Juan Universidad de Chile, Chile Universidad de , Argentina Arredondo-Ruiz, Juan Boulton, Lyonell Universidad Autonoma Metropolitana, Universidad Simon Bolivar, Venezuela Mexico Brambila, Fernando Ashbaugh, Mark Universidad Nacional Autonoma de University of Missouri, USA Mexico, Mexico

Astaburuaga, Maria Brock, Friedemann Pontificia Universidad Catolica de Universidad de Chile, Chile Chile, Chile Busca, Jerome Averbuj, Corina CNRS, CEREMADE, Universite Paris U niversidad de Buenos Aires, Argentina Dauphine, France Avila, Andres Cabrales, Roberto Universidad de la Frontera, Chile IMECC-UNICAMP, Azizieh, Celine Caffarelli, Luis Universite Libre de Bruxelles, University of Texas, USA Baia, Margarida Capdeboscq, Yves Carnegie Mellon University, USA INSA-Rennes, France Baillon, Jean Carvajal, Xavier Universite Paris I Pantheon-Sorbonne, Instituto de Matematica Pura e France Aplicada, Brazil

ix X PARTICIPANTS

Casado, Juan De Hoop, Maarten Universidad Sevilla, Colorado School of Mines, USA

Champion, Thierry Delgado, Marilyn Universite de Toulon et du Var, France Universidad Nacional de Trujillo, Peru

Chappa, Eduardo Del Pino, Manuel Texas A&M University, USA Universidad de Chile, Chile

De Napoli, Pablo Chuaqui, Martin Universidad de Buenos Aires, Argentina Pontificia U niversidad Cat6lica de Chile, Chile De Teresa, Luz Universidad Nacional Aut6noma de Cohen, Albert Mexico, Mexico Carnegie Mellon University, USA Duran, Mario Cominetti, Roberto Pontificia U niversidad Cat6lica de Universidad de Chile, Chile Chile, Chile Conca, Carlos Universidad de Chile, Chile Elgueta, Manuel Pontificia Universidad Cat6lica de Contreras, Andres Chile, Chile U niversidad de Chile, Chile Elnazarov, Addis Corcho, Adan CMM-Universidad de Chile, Chile Instituto de Matematica Pura e Aplicada, Brazil Erdmann, Grant University of Minnesota, USA Coronel, Anibal U niversidad de Concepcion, Chile Etcheverry, Javier Centro de Investigaci6n Industrial Coronel, Alvaro Universidad de Chile, Chile Felmer, Patricio Universidad de Chile, Chile Cortes, Luis Universidad del Bio Bio, Chile Fernandez, Claudio Pontificia Universidad Cat6lica de Cortes, Victor Chile, Chile Pontificia Universidad Cat6lica de Chile, Chile Fernandez, Julian Dafermos, Constantine Universidad de Buenos Aires, Argentina Brown University, USA Ferraro, Sebastian Davila, Juan Universidad Nacional del Sur, Universidad de Chile, Chile Argentina De Almeida, Joni Flores, Isabel FEEC-UNICAMP, Brazil Pontificia U niversidad Cat6lica de Chile, Chile De Figueiredo, Djairo Universidade Estadual de Campinas, Flores, Salvador Brazil Universidad de Chile, Chile PARTICIPANTS xi

Fonseca, German Gutierrez, Sergio Universidad Nacional de Colombia, Pontificia U niversidad Catolica de Colombia Chile, Chile Fonseca, Irene Hernandez, Gaston Carnegie Mellon University, USA University of Connecticut, USA Foss, Mikil Herron, Sigifredo Kansas State University, USA Universidad Nacional de Colombia, Colombia Frigyik, Bela University of Washington, USA Huentutripay, Jorge Universidad de los Lagos, Chile Friz, Luis U niversidad del Bfo-Bfo, Chile Interial, Beatrfz Universidad Nacional Autonoma de Gabetta, Ester Mexico, Mexico Pavia University, Italy Issartel, Jean Garda, Galina Ecole Nationale des Ponts et Chaussees, CMM-Universidad de Chile, Chile France Garda-Huidobro, Marta Kavian, Otared Pontificia Universidad Catolica de Universite de Versailles, France Chile, Chile Kinderlehrer, David Ghoussoub, Nassif Carnegie Mellon University, USA Pacific Institute of Mathematical Sciences, and University of British Kowalczyk, Michael Columbia, Canada Kent State University, USA Gibson, Peter Krstulovic, Giorgio University of Calgary, Canada Universidad de Chile, Chile Girao, Pedro Kuchment, Peter Instituto Superior Tecnico, Lisbon, Texas A&M University, USA Kusiak, Steven Gossez, Jean-Pierre University of Washington, USA Universite Libre de Bruxelles, Belgium Lara, Luis Groisman, Pablo Universidad Nacional de Trujillo, Peru Universidad de Buenos Aires, Argentina Lassas, Matti Grunbaum, Alberto University of Helsinki, Finland University of California, Berkeley, USA Leal, Roberto Gustafson, Stephen Universidad de la Frontera, Chile University of British Columbia, Canada Lecaros, Rodrigo Gutierrez, Cristian Universidad de Chile, Chile Temple University, USA Leitao, Antonio Gutierrez, Norma Federal University of St. Catarina, Universidad Nacional de Trujillo, Peru Brazil xii PARTICIPANTS

Li, Yanyan Mercado, Jose Rutgers University, USA Instituto Mexicano de Tecnologia del Agua, Mexico Linares, Felipe Instituto de Matematica Pura e Merdan, Huseyin Aplicada, Brazil University of Pittsburgh, USA

Lopez, Francisco Milanes, Aniura Universidad Nacional Autonoma de Universidade Estadual de Campinas, Mexico, Mexico Brazil

Lorca, Sebastian Molina, Juan Universidad de Tarapaca, Chile Universidad de Concepcion, Chile Montenegro, Marcelo Macia, Fabricio Universidade Estadual de Campinas, Ecole Normale Superieure, France Brazil Manasevich, Raul Orive, Rafael U niversidad de Chile, Chile Universidad Autonoma de Madrid, Markowich, Peter Spain and Universidad de Chile, Chile University of Vienna, Austria Ortega, Jaime Universidad de Chile, Chile Marmolejo-Olea, Emilio Universidad Nacional Autonoma de Osses, Axel Mexico, Mexico Universidad de Chile, Chile Martinez, Salome Pacella, Filomena Universidad de Chile, Chile Universita de Roma, Italy Martinez, Sandra Paivarinta, Lassi Universidad de Buenos Aires, Argentina University of Oulu, Finland

Mayorga, Juan Panthee, Mahendra Universidad de Chile, Chile Instituto de Matematica Pura e Aplicada, Brazil Mazzeo, Rafe Stanford University, USA Pareschi, Lorenzo University of Ferrara, Italy Mazzucato, Anna Perez, Salvador Yale University, USA Universidad Nacional Autonama de McLaughlin, Joyce Mexico, Mexico Rensselaer Polytechnic Institute, USA Petrelli, Luca Melrose, Richard Carnegie Mellon University, USA Massachusetts Institute of Technology, Pizarro, Claudio USA Universidad de Chile, Chile Menares, Ricardo Planas, Gabriela Universidad de Chile, Chile IMECC-UNICAMP, Brazil Mendoza, Gerardo Popovici, Cristina Temple University, USA Carnegie Mellon University, USA PARTICIPANTS xiii

Pramanik, Malabika Skokan, Michal University of Wisconsin, Madison, USA University of Washington, USA

Prat, Arturo Sweers, Guido Universidad de Chile, Chile Delft University of Technology, Holland Rachele, Lizabeth Tamasan, Alexandru University at Albany, SUNY, USA University of Toronto, Canada

Radu, Petronela Tataru, Daniel Carnegie Mellon University, USA University of California, Berkeley, USA Tataru, Grigore Ramaseshan, Karthik Massachusetts Institute of Technology, University of Washington, USA USA Rapoport, Diego Terracini, Susanna Universidad de Buenos Aires, Argentina Universita di Milano Bicocca, Italy

Reitich, Fernando Timofte, Claudia University of Minnesota, USA University of Bucharest, Rumania Rico, Juan Toro, Eleuterio Universidad Nacional Aut6noma de University of Trento, Italy Mexico, Mexico Torres, Felipe Rincon, Cesar U niversidad de Chile, Chile Universidad de los Andes, Colombia Toscani, Giuseppe Rolfsen, Dale Universita di Pavia, Italy University of British Columbia, Canada Trenado, Carlos Rossi, Julio New Mexico State University, USA Universidad de Buenos Aires, Argentina Ubilla, Pedro Ruiz, David Universidad de Santiago de Chile, Chile Universidad de Granada, Spain Uriarte-Tuero, Ignacio Yale University, USA Sa Barreto, Antonio Purdue University, USA Vanninathan, Muthusamy Tata Institute of Fundamental Research Salo, Mikko Centre, Bangalore, University of Washington Velasquez, Olivia Sandoval, Maria Universidad de la Frontera, Chile Universidad Nacional Aut6noma de Mexico, Mexico Villegas-Blas, Carlos Universidad Nacional Aut6noma de Santos, Pedro Mexico, Mexico Carnegie Mellon University, USA Vogelius, Michael Schwartz, Fernando Rutgers University, USA Universidad de Chile, Chile Weder, Ricardo Sirbu, Mihai Universidad Nacional Aut6noma de Carnegie Mellon University, USA Mexico, Mexico xiv PARTICIPANTS

Weikhard, Rudi University of Alabama, USA Wunsch, Jared Northwestern University, USA Yoon, Jeong-Rock Rensselaer Polytechnic Institute, USA Zhang, Lei Texas A&M University, USA Zhou, Gang University of Toronto, Canada Zubelli, Jorge Instituto de Matematica Pura e Aplicada, Brazil This proceedings volume is a collection of articles from the Pan-American Advanced Studies Institute on partial differential equations, nonlinear analysis and inverse problems held in Santiago (Chile). Interactions among partial differential equations, nonlinear analysis, and inverse problems have produced remarkable developments over the last couple of decades. This volume contains survey articles reflecting the work of leading experts who presented minicourses at the event. Contributors include J. Busca, Y. Capdeboscq, M.S. Vogelius, F. A. Griinbaum, L. F. Matusevich, M. de Hoop, and P. Kuchment. The volume is suitable for graduate students and researchers interested in partial differen- tial equations and their applications in nonlinear analysis and inverse problems.

ISBN 0-8218-3448-7

9 780821 834480