Calculus of Variations and Partial Differential Equations: Topics on Geometrical Evolution Problems and Degree Theory

ISBN-13: 9783540648031 / Angielski / Miękka / 2000 / 348 str.

Luigi Ambrosio; Norman Dancer; M. K. Murthy

Opis

At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to PDEs respectively. This self-contained presentation accessible to PhD students bridged the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and neatly illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.

Recenzje

"Fazit: Dies ist kein typischer Proccedingsband, sondern stellt - insbesondere wegen der Lecture Notes von Dancer und vor allem wegen der von Ambrosio - ... eine wichtige Bereicherung der anspruchsvollen Forschungsliteratur auf den Gebieten Variationsrechnung und partielle Differentialgleichungen dar. Es ist erfreulich, dass hier allerneueste Forschungsergebnisse so schnell Eingang - abgesehen von Zeitschriften - in die Literatur gefunden haben. Für Interessierte eine lohnenswerte Anschaffung!"
Jahresbericht der DMV, 104. Band, Heft 2, August 2002

Spis treści

I Geometric Evolution Problems.- Geometric evolution problems, distance function and viscosity solutions.- Variational models for phase transitions, an approach via ?-convergence.- Some aspects of De Giorgi’s barriers for geometric evolutions.- Partial Regularity for Minimizers of Free Discontinuity Problems with p-th Growth.- Free discontinuity problems and their non-local approximation.- II Degree Theory on Convex Sets and Applications to Bifurcation.- Degree theory on convex sets and applications to bifurcation.- Nonlinear elliptic equations involving critical Sobolev exponents.- On the existence and multiplicity of positive solutions for semilinear mixed and Neumann elliptic problems.- Solitons and Relativistic Dynamics.- An algebraic approach to nonstandard analysis.- References.

Blurb

The link between Calculus of Variations and Partial Differential Equations has always been strong, because variational problems produce, via their Euler-Lagrange equation, a differential equation and, conversely, a differential equation can often be studied by variational methods. At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on a classical topic (the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to pde's resp.), in a self-contained presentation accessible to PhD students, bridging the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and nicely illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.