Calculus of Variations and Geometric Evolution Problems: by Fabrice Bethuel, Gerhard Huisken, Stefan Müller, Klaus

By Fabrice Bethuel, Gerhard Huisken, Stefan Müller, Klaus Steffen (auth.), Stefan Hildebrandt, Michael Struwe (eds.)

The overseas summer time institution on Calculus of diversifications and Geometric Evolution difficulties was once held at Cetraro, Italy, 1996. The contributions to this quantity replicate relatively heavily the lectures given at Cetraro that have supplied a picture of a pretty huge box in research the place lately we've seen many very important contributions. one of the issues handled within the classes have been variational equipment for Ginzburg-Landau equations, variational versions for microstructure and section transitions, a variational therapy of the Plateau challenge for surfaces of prescribed suggest curvature in Riemannian manifolds - either from the classical viewpoint and within the environment of geometric degree theory.

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Extra resources for Calculus of Variations and Geometric Evolution Problems: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, June 15–22, 1996

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2. Variational methods Here we come back to the general case, without symmetries. In view of the previous analysis one might expect to find more solutions as d increases. consistent with the following. This hope is also P r o p o s i t i o n 17. A s s u m e f~ is starshaped, and that g = 1 on O~ ; then the only solution to GL~ is v -- 1. Proof : Use Pohazaev's identity to assert that M = 0 so that Iv I = 1 on fl, and GL~ writes Av=0. The conclusion follows. In the sequel of this section, we will therefore assume that d ~ 2, and will use Morse theory to construct solution.

H. Lin. He proved that, if the time t is scaled by Ilog r then the vortices move (in the limit r --~ 0) according to the opposite of the gradient of the renormalized energy (see also Jerrard and Soner for related results [GS]). VIII. THE SCHRODINGER EQUATION Here we assume that the domain is IR2. The Schr6dinger equation related to the Ginzburg-Landau functional i,,, = Au + u (~ - {u{ ~) u(x, 0) = on [0, + o o [ • u0(x). It appears in various models in physics, for instance superfluidity, nonlinear optics, or fluid dynamics.

What are tile multiplicities of the vortices of the non-minimizing solution constructed in Theorem 5 ? (In [AB2] it is conjectured that the solution has either a vortex of multiplicity 2, and d - 2 vortices of degree 1, or d + 1 vortices of degree +1, and one vortex of degree - 1 ) . Question 3. W h a t are the multiplicities of the vortices in Theorem 4 ? In particular are there vortices of negative charge ? Question 4. Is it possible to extend the methods of Theorem 4 (or Theorem 5) to the Abelian Higgs model on ]R2 (see [JT]) ?

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