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Self-dual Partial Differential Systems and Their Variational Principles (Hardcover, 2009 ed.)
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Self-dual Partial Differential Systems and Their Variational Principles (Hardcover, 2009 ed.)
Series: Springer Monographs in Mathematics
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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.
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