Gradient flow equations of non convex functionals

seminar: Global solutions to the "truncated" Perona-Malik equation in one dimension

speaker: Matteo Novaga (Unversità di Pisa)

abstract: We study the $L^2$-gradient flow of the nonconvex functional $\energy_\p(u) := \frac{1}{2}\int_{(0,1)} \phi(u_x)~dx$, 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 $\inidat$, i.e., when $\inidat$ is a piecewise smooth function having derivative taking values both in the region where $\p''>0$ and where $\p'' = 0$. We show that, in general, the region where the derivative of $u$ takes values where $\p'' =0$ progressively disappears while the region where $\phi''$ is positive grows. We show this behaviour with some numerical experiments.

timetable:
Mon 2 May, 11:30 - 12:30, Sala Conferenze Centro De Giorgi
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