Bücher der Reihe Progress in Nonlinear Differential Equations and Their Applications

This monograph presents a rigorous mathematical introduction to optimal transport as a variational problem, its use in modeling various phenomena, and its connections with partial differential equations.

The "Dynamical Systems Semester" took place at the Euler International Mathematical Institute in St. Petersburg, Russia, in the autumn of 1991. Since the new building of the Euler Institute was not ready at that moment, the sessions were held in the old building of the Steklov Mathemati cal Institute in the very center of St. Petersburg.

This volume presents the proceedings of a series of lectures hosted by the Math ematics Department of The University of Tennessee, Knoxville, March 22-24, 1995, under the title "Nonlinear Partial Differential Equations in Geometry and Physics" .

This self-contained research monograph focuses on semilinear Dirichlet problems and similar equations involving the p-Laplacian. The author explains new techniques in detail, and derives several numerical methods approximating the concentration point and the free boundary. The corresponding plots are highlights of this book.

This monograph has grown out of research we started in 1987, although the foun dations were laid in the 1970's when both of us were working on our doctoral theses, trying to generalize the now classic paper of Oleinik, Kalashnikov and Chzhou on nonlinear degenerate diffusion.

viii Preface Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport prob lems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework.

A collection of research articles originating from the Workshop on Nonlinear Analysis and Applications held in Bergamo in July 2001.Classical topics of nonlinear analysis were considered, such as calculus of variations, variational inequalities, critical point theory and their use in various aspects of the study of elliptic differential equations and systems, equations of Hamilton-Jacobi, Schrödinger and Navier-Stokes, and free boundary problems. Moreover, various models were focused upon: travelling waves in supported beams and plates, vortex condensation in electroweak theory, information theory, non-geometrical optics, and Dirac-Fock models for heavy atoms.

Semilinear elliptic equations play an important role in many areas of mathematics and its applications to other sciences. This book presents a wealth of modern methods to solve such equations. Readers of this exposition will be advanced students and researchers in mathematics, physics and other.

Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampere-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics.

The growing interest in these problems is reflected by the series of meetings held under the title "Free Boundary Problems: Theory and Applications" (Ox ford 1974, Pavia 1979, Durham 1978, Montecatini 1981, Maubuisson 1984, Irsee 1987, Montreal 1990, Toledo 1993, Zakopane 1995, Crete 1997, Chiba 1999).

Highlights developments in nonlinear analysis and differential equations. This work covers: periodic solutions of systems with p-Laplacian type operators; bifurcation in variational inequalities; a geometric approach to dynamical systems in the plane via twist theorems; and asymptotic behavior and periodic solutions for Navier-Stokes equations.

The growing interest in these problems is reflected by the series of meetings held under the title "Free Boundary Problems: Theory and Applications" (Ox ford 1974, Pavia 1979, Durham 1978, Montecatini 1981, Maubuisson 1984, Irsee 1987, Montreal 1990, Toledo 1993, Zakopane 1995, Crete 1997, Chiba 1999).

This volume presents lectures from the Danish-Swedish Analysis Seminar 1995, covering themes in mathematical physics, spanning a wide variety of topics.

A summary and synthesis of recent research demonstrating that variational methods can be used to successfully handle systems with singular potential, the Lagrangian systems. The classic cases of the Kepler problem and the N-body problem are used as specific examples.

The book is based on my lecture notes "Infinite dimensional Morse theory and its applications", 1985, Montreal, and one semester of graduate lectures delivered at the University of Wisconsin, Madison, 1987.

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems.

In recent years many researchers in material science have focused their attention on the study of composite materials, equilibrium of crystals and crack distribution in continua subject to loads.

This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes - non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that played a major role in Savin's proof. It also introduces an alternative method for obtaining pointwise estimates. Key features and topics of this self-contained, systematic exposition include:. Resolution of the structure of minimal solutions in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions.. Preliminary material beginning with the stress-energy tensor, via which monotonicity formulas, and Hamiltonian and Pohozaev identities are developed, including a self-contained exposition of the existence of standing and traveling waves.. Tools that allow the derivation of general properties of minimizers, without any assumptions of symmetry, such as a maximum principle or density and pointwise estimates.. Application of the general tools to equivariant solutions rendering exponential estimates, rigidity theorems and stratification results. This monograph is addressed to readers, beginning from the graduate level, with an interest in any of the following: differential equations - ordinary or partial; nonlinear analysis; the calculus of variations; the relationship of minimal surfaces to diffuse interfaces; or the applied mathematics of materials science.

This monograph presents controllability and stabilization methods in control theory that solve parabolic boundary value problems.

The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems.Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles.

Vsevolod Alekseevich Solonnikov is known as one of the outstanding mathematicians from the St Petersburg Mathematical School. This book contains twenty original contributions on many topics related to V A Solonnikov's work, selected from the invited talks of an international conference held on the occasion of his 70th birthday.

This volume has grown from a conference entitled Harmonic Maps, Minimal Sur­ faces and Geometric Flows which was held at the Universite de Bretagne Occi­ dentale from July 7th-12th, 2002, in the town of Brest in Brittany, France. We welcomed many distinguished mathematicians from around the world and a dy­ namic meeting took place, with many fruitful exchanges of ideas. In order to produce a work that would have lasting value to the mathematical community, the organisers decided to invite a small number of participants to write in-depth articles around a common theme. These articles provide a balance between introductory surveys and ones that present the newest results that lie at the frontiers of research. We thank these mathematicians, all experts in their field, for their contributions. Such meetings depend on the support of national organisations and the local community and we would like to thank the following: the Ministere de l'Education Nationale, Ministere des Affaires Etrangeres, Centre National de Recherche Sci en­ tifique (CNRS), Conseil Regional de Bretagne, Conseil General du Finistere, Com­ munaute Urbaine de Brest, Universite de Bretagne Occidentale (UBO), Faculte des Sciences de l'UBO, Laboratoire de Mathematiques de l'UBO and the Departement de Mathematiques de l'UBO. Their support was generous and ensured the success of the meeting. We would also like to thank the members of the scientific committee for their advice and for their participation in the conception and composition of this volume: Pierre Berard, Jean-Pierre Bourguignon, Frederic Helein, Seiki Nishikawa and Franz Pedit.

These Proceedings comprise the bulk of the papers presented at the Inter national Conference on Semigroups of Opemtors: Theory and Contro~ held 14-18 December 1998, Newport Beach, California, U.S.A.

This book presents a hierarchy of macroscopic models for semiconductor devices, studying three classes of models in detail: isentropic drift-diffusion equations, energy-transport models, and quantum hydrodynamic equations.

The international conference on which the book is based brought together many of the world's leading experts, with particular effort on the interaction between established scientists and emerging young promising researchers, as well as on the interaction of pure and applied mathematics.