A Boundary Control Problem for a Nonlinear Parabolic by Maksimov V. I.

By Maksimov V. I.

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Extra info for A Boundary Control Problem for a Nonlinear Parabolic Equation

Example text

P) arc called the principal curvatures of F(34") at F(p). Also note that at a given point p 9 34n by choosing normal coordinates and then possibly rotating them we can always arrange that at this point gij = ~ij, 7"e / = O, Ve, - hi~ = h ii = diag(Al,... A,~). , the Gauss-Kroncckcr curvaturc by K := d e t ( W ) = det{h}} - det(h,~} det{gi~} - ,~l . . . )~,,, the total curvature by ]A[2 : = t r ( W t W ) , ~ = h i i h q = g ik g i, hilh~a = ,~ + . . + ~ , = ttjl~ and the scalar curvature (ill Euclidean space 1R"+l) by R = II 2 - ]A[ ~ = 2(~1A2 + )~Aa + " " + )~,-~)~,,).

In the sequel of this section, we will therefore assume that d ~ 2, and will use Morse theory to construct solution. At this point, only a few results have been obtained. We will sketch the proof of the following theorem (see Almeida-B [AB2]). T h e o r e m 5. If ~ is sumciently small, then (GL~) has at least three distinct solutions, among which one at least is non-minimizing. Remark. H. Lin [Lil]. For special b o u n d a r y conditions g, he was able to produce solutions with vortices of opposite sign, which are local-minimizers, using heat flow methods.

Back to Ginzburg-Landau 27 We are going to show how the previous analysis connects to Ginzburg-Landau functionals, and Theorem 1 : we will sketch the proof of the fact that the configuration ( a l , . . , ad) in Theorem 1 has to be minimizing for the renormalized energy. As seen in Theorem 1, any sequence u , , converges, up to a subsequence, to a map u. of the form d z -- a i u, = H ~ e x p iV i=l where ~p is harmonic. JT=, {a,}. For instance for auy compact subset g of a \ UT=, {a,}, we have (31) ll~, - ~,IIc,(i<) < cK ~2, where CK depends on K.

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