Items related to Self-dual Partial Differential Systems and Their Variational...
Synopsis
How to solve partial differential systems by completing the square This could well have been the title of this monograph as it grew into a project to develop a s tematic approach for associating suitable nonnegative energy functionals to a large class of partial differential equations PDEs and evolutionary systems The minima of these functionals are to be the solutions we seek not because they are critical points i e from the corresponding Euler Lagrange equations but from also ing zeros of these functionals The approach can be traced back to Bogomolnyi s trick of completing squares in the basic equations of quantum eld theory e g Yang Mills Seiberg Witten Ginzburg Landau etc which allows for the deri tion of the so called self or antiself dual version of these equations In reality the self dual Lagrangians we consider here were inspired by a variational proach proposed over 30 years ago by Brezis and Ekeland for the heat equation and other gradient ows of convex energies It is based on Fenchel Legendre ality and can be used on any convex functional not just quadratic ones making them applicable in a wide range of problems In retrospect we realized that the ergy identities satis ed by Leray s solutions for the Navier Stokes equations are also another manifestation of the concept of self duality in the context of evolution equations The classic cycle routes for two of the nation s top biking destinations Cycling routes that range from family friendly paved bicycle paths to epic mountain pass climbs Easy reference lists such as Best Family Friendly Rides Top Climbing Challenges Best Water Views and more Full color with maps photographs and lively turn by turn descriptionsFrom a 50 mile route around Orcas Island to an after work ride through a Portland neighborhood these new cycling guidebooks offer truly classic routes for two of the nation s most bike crazy states Whether you re soaring down a Cascade s pa
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