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Constructing generic smooth maps of a manifold into a surface with prescribed singular loci

Osamu Saeki (1995)

Annales de l'institut Fourier

It is known that the singular set S ( f ) of a generic smooth map f : M N of an n -dimensional manifold into a surface is a closed 1-dimensional submanifold of M and that it has a natural stratification induced by the absolute index. In this paper, we give a complete characterization of those 1-dimensional (stratified) submanifolds which arise as the singular set of a generic map in terms of the homology class they represent.

Continuity of hysteresis operators in Sobolev spaces

Pavel Krejčí, Vladimír Lovicar (1990)

Aplikace matematiky

We prove that the classical Prandtl, Ishlinskii and Preisach hysteresis operators are continuous in Sobolev spaces W 1 , p ( 0 , T ) for 1 p < + , (localy) Lipschitz continuous in W 1 , 1 ( 0 , T ) and discontinuous in W 1 , ( 0 , T ) for arbitrary T > 0 . Examples show that this result is optimal.

Continuous dependence for solution classes of Euler-Lagrange equations generated by linear growth energies

Ken Shirakawa (2009)

Banach Center Publications

In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

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