By Bethuel, Huisken, Müller and Steffen
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Additional info for Calculus of Variations and Geometric Evolution Problems
V = O f o r i 1, @Q 9 ~ d, D . v = 0 on O~ v • v=, + ~5~2). e. vxv~ =---H~ 2 + f f ~ t , v x v ~ t = H ~ t-':xbx 2. 25 We deduce that f~. IWl~ = fn, I~ x Wl ~ (30) -- f.. fVr = + f.. IVHI 2 +2/,. H==r H,:, r = L Iv+12+lvHI2' Indeed Since D - u = 0 on 0~, H~ = 0 on 0~. Integrating by parts we have fo ~H" = ~o ~ r H = O (since ~ = Cte on Ow,). This yields (30). We have already proved 7e >- [ [V~] 2. Next consider u = v exp iH (we take H = 0 on 0~). A simple computation shows that f~, Ivut 2 = fn.
Many problems remain open, as existence, motion low for vortices... C. Saut [BS], we have studied the existence problem for travelling wave solutions of the form. These solutions have the form U ( ~ l , X , , t ) = ~(~, - a , x ~ ) where (xl,x2) are cartesian coordinates on IR2, v is a function on IR2, and c > 0 is the speed of the wave. The equation for v reads av icg;7~, = A~ + v (1 - I~1~). 39 We establish the existence of a solution for small speeds. These solutions have been studied on a more formal level in a serie of papers (see for instance for references, Jones, P u t t e r m a n [OPrq, or Pismen and Nepomnyashchy [eN D.
Similarly we get o__: _ at = o]o o] o gl,} = g,,g (g'kl,kj) 0/ + 01 0, -2:h, = L)~V, V i f _ f Of (h,kh~ + ['Qo'j), Oh~j as required. The curvature terms in this nonlinear reaction-diffusion system provide the key for understanding the interaction between geometric properties of the hypersurface and the ambient manifold. They are the tool to study these geometric phenomena with analytical means. For some choices of f we will now describe recent developments. 2) exists on a maximal time interval [0,T[, 0 < T < co.
Calculus of Variations and Geometric Evolution Problems by Bethuel, Huisken, Müller and Steffen