After publishing an introduction to the Navier-Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with another set of lecture notes based on a graduate course in two parts, as indicated by the title. A draft has been available on the internet for a few years. The author has now revised and polished it into a text accessible to a larger audience.
After publishing an introduction to the Navier-Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with another set of lec...
The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements. This volume provides an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to hyperbolic conservation laws. The captivating volume contains surveys of recent deep results and provides an overview of further developments and related open problems. Readers should have basic knowledge of PDE and measure theory.
The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements. This volume provides an up-...
In the spring of 1999, I taught (at CARNEGIEMELLON University) a graduate course entitled Partial Di?erential Equations Models in Oceanography, and I wrote lecture notes which I distributed to the students; these notes were then made available on the Internet, and they were distributed to the participants of a Summer School held in Lisbon, Portugal, in July 1999. After a few years, I feel it will be useful to make the text available to a larger audience by publishing a revised version. To an uninformed observer, it may seem that there is more interest in the Navier-Stokes equation nowadays,...
In the spring of 1999, I taught (at CARNEGIEMELLON University) a graduate course entitled Partial Di?erential Equations Models in Oceanography, and I ...
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the "tropical" approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry....
The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non...
This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hormander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by...
This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this ric...
In these notes we consider two kinds of nonlinear evolution problems of von Karman type on Euclidean spaces of arbitrary even dimension. Each of these problems consists of a system that results from the coupling of two highly nonlinear partial differential equations, one hyperbolic or parabolic and the other elliptic. These systems take their name from a formal analogy with the von Karman equations in the theory of elasticity in two dimensional space. We establish local (respectively global) results for strong (resp., weak) solutions of these problems and corresponding well-posedness...
In these notes we consider two kinds of nonlinear evolution problems of von Karman type on Euclidean spaces of arbitrary even dimension. Each of th...
Providing an introduction to both classical and modern techniques in projective algebraic geometry, this monograph treats the geometrical properties of varieties embedded in projective spaces, their secant and tangent lines, the behavior of tangent linear spaces, the algebro-geometric and topological obstructions to their embedding into smaller projective spaces, and the classification of extremal cases. It also provides a solution of Hartshorne's Conjecture on Complete Intersections for the class of quadratic manifolds and new short proofs of previously known results, using the modern...
Providing an introduction to both classical and modern techniques in projective algebraic geometry, this monograph treats the geometrical propertie...
These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal...
These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short...
Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrodinger equations are given.
Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple p...
