Calculus of Variations and Partial Differential Equations de Luigi Ambrosio

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

Luigi Ambrosio

Giuseppe Buttazzo

Norman Dancer

Antonio Marino

M.K.V. Murthy

en Limba Engleză Paperback – 24 ian 2000

Calculus of variations is a major branch of analysis, and partial differential equations are used to model natural phenomena and are used in all of the physical sciences. Using degree theory and the geometric problem of evolution of a surface, this text brings these two fields together.

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Cuprins

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.

Textul de pe ultima copertă

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.