594
Recent Trends in NonlinearPartial Differential Equations I:
Evolution Problems
Workshop in Honor of Patrizia Pucci’s 60th BirthdayNonlinear Partial Differential Equations
May 28–June 1, 2012University of Perugia, Perugia, Italy
James B. SerrinEnzo L. Mitidieri
Vicentiu D. RadulescuEditors
American Mathematical Society
Recent Trends in NonlinearPartial Differential Equations I:
Evolution ProblemsWorkshop in Honor of Patrizia Pucci’s 60th Birthday
Nonlinear Partial Differential EquationsMay 28–June 1, 2012
University of Perugia, Perugia, Italy
James B. SerrinEnzo L. Mitidieri
Vicentiu D. RadulescuEditors
594
Recent Trends in NonlinearPartial Differential Equations I:
Evolution ProblemsWorkshop in Honor of Patrizia Pucci’s 60th Birthday
Nonlinear Partial Differential EquationsMay 28–June 1, 2012
University of Perugia, Perugia, Italy
James B. SerrinEnzo L. Mitidieri
Vicentiu D. RadulescuEditors
American Mathematical SocietyProvidence, Rhode Island
EDITORIAL COMMITTEE
Dennis DeTurck, managing editor
Michael Loss Kailash Misra Martin J. Strauss
2010 Mathematics Subject Classification. Primary 35-06; Secondary 34-06, 35K55,35L70, 47-06, 58-06.
Frontispiece photo taken by Professor Michel Chipot during a visit by Professor Pucci tothe University in Zurich
Library of Congress Cataloging-in-Publication Data
Workshop on nonlinear partial differential equations (2012 : Perugia, Italy)Recent trends in nonlinear partial differential equations : a workshop on nonlinear partial differ-ential equations : in honor of Patrizia Pucci’s 60th birthday, May 28–June 1, 2012, University ofPerugia, Perugia, Italy / James B. Serrin, Enzo L. Mitidieri, Vicentiu D. Radulescu, editors.
2 volumes ; cm – (Contemporary Mathematics ; volume 594–595)Includes bibliographical references.ISBN 978-0-8218-8736-3 (alk. paper) – ISBN 978-0-8218-9861-1 (alk. paper)1. Differential equations, Nonlinear-Congresses. 2. Differential equations, Parabolic–Congresses.
I. Pucci, Patrizia, honouree. II. Serrin, J. (James), 1926–2012, editor of compilation. III. Miti-dieri, Enzo, editor of compilation. IV. Radulescu, Vicentiu D., 1958– editor of compilation. V.Title.
QA372.W82 2012515′.353–dc23 2013017799
Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)
DOI: http://dx.doi.org/10.1090/conm/594
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10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13
Contents
Thanks vii
Preface ix
A tribute to Patrizia PucciJean Mawhin 1
A scientific profile of Patrizia PucciGianluca Vinti 9
The Liouville theorem for homogeneous elliptic differential inequalitiesJames B. Serrin 19
Multiplicity results for porous media and fast diffusion equations with aquadratic gradient term
Boumediene Abdellaoui, Ireneo Peral,
and Magdalena Walias 37
A priori estimates for solutions of p–Kirchhoff systems under dynamicboundary conditions
Giuseppina Autuori 59
Higher order linear parabolic equationsGerassimos Barbatis and Filippo Gazzola 77
Sources of nonlinearity in the kinetic theory for active particles with focus onthe formation of political opinions
Nicola Bellomo and Vincenzo Coscia 99
Yamabe type equations with sign-changing nonlinearities on the Heisenberggroup, and the role of Green functions
Bruno Bianchini, Luciano Mari, and Marco Rigoli 115
Asymptotic behavior of the solution to variational inequalities with jointconstraints on its value and its gradient
Michel Chipot and Karen Yeressian 137
Existence and uniqueness of solutions to a Cauchy problem modeling thedynamics of socio-political conflicts
Francesca Colasuonno and Maria Cesarina Salvatori 155
Regularity points of ACL–homeomorphisms in the planeFernando Farroni, Raffaella Giova, and Carlo Sbordone 167
v
vi CONTENTS
Energy estimates and integral Harnack inequality for some doubly nonlinearsingular parabolic equations
Simona Fornaro, Maria Sosio, and Vincenzo Vespri 179
Initial value problems for diffusion equations with singular potentialKonstantinos T. Gkikas and Laurent Veron 201
On a class of fully nonlinear PDEs from complex geometryErmanno Lanconelli and Annamaria Montanari 231
Concentration of conformal volume, improved Moser-Trudinger inequalitiesand Toda systems
Andrea Malchiodi 243
Periodic solutions of Lagrangian difference systems: Periodic nonlinearities(almost) don’t matter
Jean Mawhin 265
A limit problem for degenerate quasilinear variational inequalities in cylindersDimitri Mugnai 281
Strong solutions for the wave equation with a kinetic boundary conditionEnzo Vitillaro 295
Thanks
I wish to express my deepest thanks to Enzo Mitidieri and Vicentiu Radulescufor their immeasurable efforts and dedication in bringing to print these volumes ofcollected papers from the talks given at the May 2012 conference held in Perugiato print. I also want to convey to the authors of these works my sincerest thanksfor dedicating their excellent research and profound scientific contributions to theseproceedings, including special thanks again to the speakers and the organizers whocontributed so much to the high level of the conference in Perugia.
Regrettably, I cannot thank James Serrin in person for all his contributionsto both the Perugia conference and the publication of these volumes, as he passedaway in August of this year before it could be published. He was an enthusiasticorganizer of the May conference and played a crucial role as editor of the collectionwhich he was very much looking forward to seeing published. Even though atthe time of the conference he was already not well, he attended all the lecturesand delivered an exceptional and unforgettable opening lecture. His presence inPerugia at the conference was for me the most appreciated gift I received for my60th birthday.
James was a great mathematician and an exceptional man. His work has in-spired generations of mathematicians and will continue to influence the future ofmathematics. His loss is inconsolable for anyone who had the good fortune to meethim as I had.
Patrizia PucciPerugia, December 2012
vii
Preface
On May 11, 2012, Professor Patrizia Pucci turned 60. In celebration of PatriziaPucci’s birthday, a conference was held at the University of Perugia during May 28and June 1, 2012. The main purpose of this scientific meeting has been to bringtogether leading experts and researchers in nonlinear partial differential equations,to promote research and to stimulate interactions among the participants. Anatmosphere of particular warmth and the high scientific level made this conferencememorable. The program testified to the wide ranging influence of Professor Puccion the field of nonlinear analysis and partial differential equations. Many of thespeakers also spoke about the generosity of Patrizia in welcoming and encouragingyoung new researchers.
In her own work, Patrizia Pucci has been a seminal influence in many importantareas: the maximum principle, qualitative analysis of solutions to many classesof nonlinear PDEs (Kirchhoff problems, polyharmonic systems), mountain passtheorem in the critical case, critical exponents, variational identities, as well asvarious degenerate or singular phenomena in mathematical physics. This samebreadth is reflected in the mathematical papers included in these two volumes.
The articles in these volumes are primarily from the conference held in Perugia.They present an original view of the state of the art of many aspects of nonlinearpartial differential equations. The articles, written by mathematicians at the centerof current developments, provide somewhat more personal views of the importantdevelopments and challenges. The first volume contains contributions on the fieldof nonlinear evolution partial differential equations. The first contribution to thisvolume is a typed version of the slides Professor James Serrin used in the openingconference of the workshop.
We are delighted that these two volumes appear in the prestigious Contempo-rary Mathematics Series of the American Mathematical Society. We are gratefulto Sergei Gelfand, the AMS Book Program Publisher, who gave his wholeheartedsupport to this project. We address special thanks to Christine Thivierge for herprofessional guidance throughout the publication process.
Enzo L. MitidieriVicentiu D. Radulescu
ix
Selected Published Titles in This Series
595 James B. Serrin, Enzo L. Mitidieri, and Vicentiu D. Radulescu, Editors, RecentTrends in Nonlinear Partial Differential Equations II, 2013
594 James B. Serrin, Enzo L. Mitidieri, and Vicentiu D. Radulescu, Editors, RecentTrends in Nonlinear Partial Differential Equations I, 2013
593 Anton Dzhamay, Kenichi Maruno, and Virgil U. Pierce, Editors, Algebraic andGeometric Aspects of Integrable Systems and Random Matrices, 2013
592 Arkady Berenstein and Vladimir Retakh, Editors, Noncommutative BirationalGeometry, Representations and Combinatorics, 2013
591 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, VladimirMaz’ya, Simeon Reich, David Shoikhet, Gilbert Weinstein, and LawrenceZalcman, Editors, Complex Analysis and Dynamical Systems V, 2013
590 Ursula Hamenstadt, Alan W. Reid, Rubı Rodrıguez, Steffen Rohde,and Michael Wolf, Editors, In the Tradition of Ahlfors-Bers, VI, 2013
589 Erwan Brugalle, Maria Angelica Cueto, Alicia Dickenstein, Eva-Maria
Feichtner, and Ilia Itenberg, Editors, Algebraic and Combinatorial Aspects ofTropical Geometry, 2013
588 David A. Bader, Henning Meyerhenke, Peter Sanders, and Dorothea Wagner,Editors, Graph Partitioning and Graph Clustering, 2013
587 Wai Kiu Chan, Lenny Fukshansky, Rainer Schulze-Pillot, and Jeffrey D.Vaaler, Editors, Diophantine Methods, Lattices, and Arithmetic Theory of QuadraticForms, 2013
586 Jichun Li, Hongtao Yang, and Eric Machorro, Editors, Recent Advances inScientific Computing and Applications, 2013
585 Nicolas Andruskiewitsch, Juan Cuadra, and Blas Torrecillas, Editors, HopfAlgebras and Tensor Categories, 2013
584 Clara L. Aldana, Maxim Braverman, Bruno Iochum, and Carolina NeiraJimenez, Editors, Analysis, Geometry and Quantum Field Theory, 2012
583 Sam Evens, Michael Gekhtman, Brian C. Hall, Xiaobo Liu, and Claudia Polini,Editors, Mathematical Aspects of Quantization, 2012
582 Benjamin Fine, Delaram Kahrobaei, and Gerhard Rosenberger, Editors,Computational and Combinatorial Group Theory and Cryptography, 2012
581 Andrea R. Nahmod, Christopher D. Sogge, Xiaoyi Zhang, and Shijun Zheng,Editors, Recent Advances in Harmonic Analysis and Partial Differential Equations, 2012
580 Chris Athorne, Diane Maclagan, and Ian Strachan, Editors, Tropical Geometryand Integrable Systems, 2012
579 Michel Lavrauw, Gary L. Mullen, Svetla Nikova, Daniel Panario, and LeoStorme, Editors, Theory and Applications of Finite Fields, 2012
578 G. Lopez Lagomasino, Recent Advances in Orthogonal Polynomials, Special Functions,and Their Applications, 2012
577 Habib Ammari, Yves Capdeboscq, and Hyeonbae Kang, Editors, Multi-Scaleand High-Contrast PDE, 2012
576 Lutz Strungmann, Manfred Droste, Laszlo Fuchs, and Katrin Tent, Editors,Groups and Model Theory, 2012
575 Yunping Jiang and Sudeb Mitra, Editors, Quasiconformal Mappings, RiemannSurfaces, and Teichmuller Spaces, 2012
574 Yves Aubry, Christophe Ritzenthaler, and Alexey Zykin, Editors, Arithmetic,Geometry, Cryptography and Coding Theory, 2012
573 Francis Bonahon, Robert L. Devaney, Frederick P. Gardiner, and DragomirSaric, Editors, Conformal Dynamics and Hyperbolic Geometry, 2012
For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.
This book is the first of two volumes which contain the proceedings of the Workshop onNonlinear Partial Differential Equations, held from May 28–June 1, 2012, at the Universityof Perugia in honor of Patrizia Pucci’s 60th birthday.
The workshop brought together leading experts and researchers in nonlinear partial dif-ferential equations to promote research and to stimulate interactions among the partici-pants. The workshop program testified to the wide ranging influence of Professor Pucci onthe field of nonlinear analysis and partial differential equations.
In her own work, Patrizia Pucci has been a seminal influence in many important areas:the maximum principle, qualitative analysis of solutions to many classes of nonlinear PDEs(Kirchhoff problems, polyharmonic systems), mountain pass theorem in the critical case,critical exponents, variational identities, as well as various degenerate or singular phenom-ena in mathematical physics. This same breadth is reflected in the mathematical papersincluded in this volume.
The companion volume (Contemporary Mathematics, Volume 595) is devoted to sta-tionary problems in nonlinear partial differential equations.
ISBN978-0-8218-8736-3
9 780821 887363
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