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

The foreign summer season institution on Calculus of adaptations and Geometric Evolution difficulties used to be 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 reasonably large box in research the place lately we've seen many very important contributions. one of the subject matters taken care of within the classes have been variational tools for Ginzburg-Landau equations, variational versions for microstructure and part transitions, a variational remedy of the Plateau challenge for surfaces of prescribed suggest curvature in Riemannian manifolds - either from the classical perspective and within the surroundings of geometric degree theory.

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REFERENCES [A] L. Almeida, Thesis. [AB1] L. Ahneida and F. Bethuel, Multiplicity results for the Ginzburg-Landau equation in presence of symmetries, to appear in Houston J. of Math. [AB2] L. Almeida and F. Bethuel, Topological methods for the Ginzburg-Landau equation, preprint. [BBH] F. Bethuel, H. Brezis and F. Hdlein, Ginzburg-Landau vortices, Birkha/iser, (1994). [BBH2] F. Bethuel, H. Brezis and F. H61ein, Asymptotics for the minimization of a Ginzburg-Landau functional, CMc. Var. and PDE, 1, (1993) 123-148.

1) is a nonlinear parabolic system of second order. Although there are some similarities to the harmonic map heatflow, this deformation law is more nonlinear in nature since the leading second order operator depends on the geometry of the solution at each time rather than the initial geometry. There is a very direct interplay between geometric properties of the underlying manifold (N"+l,~) and the geometry of tile evolving hypersurface which leads to applications both in differential geometry and mathematical physics.

More generally, the elementary symmetric functions - f = S,~, 1 < m _< n, satisfy -(Of/O)~i) > 0 on the convex cone F,,, = {)~ E lR'*lSl(A ) > 0, 1 < l < m}, yielding shorttime existence for corresponding initial data. 1) on Fk. In particular, this yields a shorttime existence result for the harmonic mean curvature flow on convex initial data, since - f = [t = S,,/S,,-t. iv) The inverse mean curvature flow with f = H -~ satisfies -(0f/0,k~) = H -2, yielding shorttime existence of a classical smooth solution for any initial data of positive mean curvature.