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On the gradient flow for the anisotropic area functional
Author(s) -
Pozzi Paola
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010043
Subject(s) - balanced flow , mathematics , mean curvature flow , anisotropy , flow (mathematics) , curvature , first variation , mean curvature , function (biology) , mathematical analysis , space (punctuation) , banach space , function space , geometry , computer science , physics , quantum mechanics , evolutionary biology , biology , operating system
We consider the anisotropic mean curvature flow for parametrized hypersurfaces and provide a new definition of gradient flow that takes into account the anisotropic nature of space. With this new approach we succeed in identifying the natural candidate for the anisotropic mean curvature vector using a variational method. Under the obtained flow Wulff shapes shrink self similarly. The new definition of gradient flow relies on understanding which Banach structure plays a fundamental role and performing a consistent choice of function spaces. The geometric setting we introduce also allows us to provide and justify a natural formulation for the anisotropic Willmore functional. Its first variation is computed.