Abstract. We study the L2-gradient flow of the nonconvex functional F(u):=∫(0,1)f(ux)dx, where f(ξ):=min(ξ2,1). We show the existence of a global in time possibly discontinuous solution starting from a mixed-type initial datum, i.e., when a piecewise smooth function having derivative taking values both in the region where the second derivative of f is strictly positive (so called "good region") and where it is zero (so called "bad region"). We show that, in general, the bad region progressively disappears while the good region grows. We show this behaviour with numerical experiments.

Global solutions to the gradient flow equation of a nonconvex functional / G. BELLETTINI; M. NOVAGA; E. PAOLINI. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - STAMPA. - 37:(2005), pp. 1657-1687. [10.1137/050625333]

Global solutions to the gradient flow equation of a nonconvex functional

PAOLINI, EMANUELE
2005

Abstract

Abstract. We study the L2-gradient flow of the nonconvex functional F(u):=∫(0,1)f(ux)dx, where f(ξ):=min(ξ2,1). We show the existence of a global in time possibly discontinuous solution starting from a mixed-type initial datum, i.e., when a piecewise smooth function having derivative taking values both in the region where the second derivative of f is strictly positive (so called "good region") and where it is zero (so called "bad region"). We show that, in general, the bad region progressively disappears while the good region grows. We show this behaviour with numerical experiments.
2005
37
1657
1687
G. BELLETTINI; M. NOVAGA; E. PAOLINI
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/255152
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