By Nikolai A. Shirokov

This examine monograph matters the Nevanlinna factorization of analytic services soft, in a feeling, as much as the boundary. The unusual homes of one of these factorization are investigated for the most typical periods of Lipschitz-like analytic features. The booklet units out to create a passable factorization conception as exists for Hardy sessions. The reader will locate, between different issues, the theory on smoothness for the outer a part of a functionality, the generalization of the theory of V.P. Havin and F.A. Shamoyan additionally identified within the mathematical lore because the unpublished Carleson-Jacobs theorem, the full description of the zero-set of analytic features non-stop as much as the boundary, generalizing the classical Carleson-Beurling theorem, and the constitution of closed beliefs within the new wide selection of Banach algebras of analytic services. the 1st 3 chapters suppose the reader has taken a typical path on one advanced variable; the fourth bankruptcy calls for supplementary papers mentioned there. The monograph addresses either ultimate 12 months scholars and doctoral scholars starting to paintings during this zone, and researchers who will locate the following new effects, proofs and strategies.

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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.