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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.