Calculus of Variations and Geometric Measure Theory

G. Bellettini - M. Novaga - E. Paolini

Global solutions to the gradient flow equation of a nonconvex functional

created by novaga on 03 Feb 2005
modified by paolini on 06 Feb 2008


Published Paper

Inserted: 3 feb 2005
Last Updated: 6 feb 2008

Journal: SIAM J. on Math. Anal.
Volume: 37
Number: 5
Pages: 1657-1687
Year: 2005


We study the $L^2$-gradient flow of the nonconvex functional $F(u) := \int_{(0,1)} f(u_x) dx$, where $f(\xi) := \min(\xi^2, 1)$. 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.