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Variational Methods for Discontinuous Structures
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. At the same time several new issues in computer vision and image processing have been studied in depth. The understanding of many of these problems has made significant progress thanks to new methods developed in calculus of variations, geometric measure theory and partial differential equations. In particular, new technical tools have been introduced and successfully applied. For example, in order to describe the geometrical complexity of unknown patterns, a new class of problems in calculus of variations has been introduced together with a suitable functional setting: the free-discontinuity problems and the special BV and BH functions. The conference held at Villa Olmo on Lake Como in September 1994 spawned successful discussion of these topics among mathematicians, experts in computer science and material scientists.
A Pure Soul
This biography illuminates the life of Ennio De Giorgi, a mathematical genius in parallel with John Nash, the Nobel Prize Winner and protagonist of A Beautiful Mind. Beginning with his childhood and early years of research, into his solution of the 19th problem of Hilbert and his professorship, this book pushes beyond De Giorgi’s rich contributions to the mathematics community, to present his work in human rights, including involvement in the fight for Leonid Plyushch’s freedom and the defense of dissident Uruguayan mathematician José Luis Massera. Considered by many to be the greatest Italian analyst of the twentieth century, De Giorgi is described in this volume in full through documents and direct interviews with friends, family, colleagues, and former students.
Geometric Control Theory and Sub-Riemannian Geometry
Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.
Partial Differential Equations and Mathematical Physics
The 17 invited research articles in this volume, all written by leading experts in their respective fields, are dedicated to the great French mathematician Jean Leray. A wide range of topics with significant new results---detailed proofs---are presented in the areas of partial differential equations, complex analysis, and mathematical physics. Key subjects are: * Treated from the mathematical physics viewpoint: nonlinear stability of an expanding universe, the compressible Euler equation, spin groups and the Leray--Maslov index, * Linked to the Cauchy problem: an intermediate case between effective hyperbolicity and the Levi condition, global Cauchy--Kowalewski theorem in some Gevrey classes...
Advances in the Theory of Shock Waves
In the field known as "the mathematical theory of shock waves," very exciting and unexpected developments have occurred in the last few years. Joel Smoller and Blake Temple have established classes of shock wave solutions to the Einstein Euler equations of general relativity; indeed, the mathematical and physical con sequences of these examples constitute a whole new area of research. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an ap proach had for a long time seemed out of reach. The stabili...
Mean Curvature Flow and Isoperimetric Inequalities
Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.
Maximal Function Methods for Sobolev Spaces
This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy's inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the p p-Laplace equation and the use of maximal function techniques is this context are discussed. The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.
Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients
View the abstract: https://bookstore.ams.org/memo-269-1311/
Recent Advances in Partial Differential Equations and Applications
This volume contains the proceedings of the International Conference on Recent Advances in PDEs and Applications, in honor of Hugo Beirão da Veiga's 70th birthday, held from February 17–21, 2014, in Levico Terme, Italy. The conference brought together leading experts and researchers in nonlinear partial differential equations to promote research and to stimulate interactions among the participants. The workshop program testified to the wide-ranging influence of Hugo Beirão da Veiga on the field of partial differential equations, in particular those related to fluid dynamics. In his own work, da Veiga has been a seminal influence in many important areas: Navier-Stokes equations, Stokes systems, non-Newtonian fluids, Euler equations, regularity of solutions, perturbation theory, vorticity phenomena, and nonlinear potential theory, as well as various degenerate or singular models in mathematical physics. This same breadth is reflected in the mathematical papers included in this volume.
Nonlinear Analysis and its Applications to Differential Equations
This work, consisting of expository articles as well as research papers, highlights recent developments in nonlinear analysis and differential equations. The material is largely an outgrowth of autumn school courses and seminars held at the University of Lisbon and has been thoroughly refereed. Several topics in ordinary differential equations and partial differential equations are the focus of key articles, including: * periodic solutions of systems with p-Laplacian type operators (J. Mawhin) * bifurcation in variational inequalities (K. Schmitt) * a geometric approach to dynamical systems in the plane via twist theorems (R. Ortega) * asymptotic behavior and periodic solutions for Navier--Stokes equations (E. Feireisl) * mechanics on Riemannian manifolds (W. Oliva) * techniques of lower and upper solutions for ODEs (C. De Coster and P. Habets) A number of related subjects dealing with properties of solutions, e.g., bifurcations, symmetries, nonlinear oscillations, are treated in other articles. This volume reflects rich and varied fields of research and will be a useful resource for mathematicians and graduate students in the ODE and PDE community.
