We study the L2-gradient flow of the nonconvex functional F phi(u) := 1/2 integral((0,1)) phi(u(x)) dx, where phi(xi) := min(xi(2), 1). We show the existence of a global in time possibly discontinuous solution u starting from a mixed-type initial datum u(0), i. e., when u(0) is a piecewise smooth function having derivative taking values both in the region where phi'' > 0 and where phi'' = 0. We show that, in general, the region where the derivative of u takes values where phi'' = 0 progressively disappears while the region where phi'' is positive grows. We show this behavior with some numerical experiments.

Bellettini, G., Novaga, M., Paolini, E. (2006). Global solutions to the gradient flow equation of a nonconvex functional. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 37(5), 1657-1687 [10.1137/050625333].

Global solutions to the gradient flow equation of a nonconvex functional

BELLETTINI, GIOVANNI;
2006-01-01

Abstract

We study the L2-gradient flow of the nonconvex functional F phi(u) := 1/2 integral((0,1)) phi(u(x)) dx, where phi(xi) := min(xi(2), 1). We show the existence of a global in time possibly discontinuous solution u starting from a mixed-type initial datum u(0), i. e., when u(0) is a piecewise smooth function having derivative taking values both in the region where phi'' > 0 and where phi'' = 0. We show that, in general, the region where the derivative of u takes values where phi'' = 0 progressively disappears while the region where phi'' is positive grows. We show this behavior with some numerical experiments.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - Analisi Matematica
English
Finite element method; Forward-backward parabolic equations; Nonconvex functionals
Bellettini, G., Novaga, M., Paolini, E. (2006). Global solutions to the gradient flow equation of a nonconvex functional. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 37(5), 1657-1687 [10.1137/050625333].
Bellettini, G; Novaga, M; Paolini, E
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/37220
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