By Bethuel, Huisken, Müller and Steffen

ISBN-10: 3540659773

ISBN-13: 9783540659778

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**Additional resources for Calculus of Variations and Geometric Evolution Problems**

**Example text**

Remark : Recently, in a joint work with B. Helffer, we have been able to show that the estimate in Proposition 15 is in some sense optimal. More precisely, we proved (see [BH]) that ttiere is some constant #1 > 0 such that the Morse Index of v, is less than ~lldl 2. 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.

Notice that - A f - f(lAI 2 + l~ic(~, ~)) = J f is the Jacobi operator acting on f , as is wellknown from the second variation formula for the area. P r o o f . A4'* and {y~} near F(p) in N. Arranging coordinates at a fixed point p such that g~j(p) = ~,j, (O/Ox')gjk(p) = O, ~,a(F(p)) = ~,,a, (O/Oy~')ga6(F(P)) = 0 all identities are straightforward consequences of the definitions and the Gauss-Weingarten relations. The computations have been carried out in detail for f = - U in [34], [35] and [4].

Some words on physics are in order. At low temperature, some material exhibits very special properties : they lose electric resistivity, and repel magnetic fluxes. This phenomenon is termcd superconductivity. It turns out (according to the theory developped by Bardeen, Schaeffer and Cooper) that the electric current is not mediated by isolated electrons (as in usual conductors), but by pairs of electrons (with opposite sign), which behave llke bosons. On a macroscopic level, these pairs of electrons 40 (called Cooper pairs) are modelled by a complex-valued wave function u.

### Calculus of Variations and Geometric Evolution Problems by Bethuel, Huisken, Müller and Steffen

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