Calculus of Variations and Geometric Measure Theory
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F. Fleißner - G. Savaré

Reverse approximation of gradient flows as Minimizing Movements: a conjecture by De Giorgi

created by savare on 22 Nov 2017
modified on 23 Nov 2017

[BibTeX]

Submitted Paper

Inserted: 22 nov 2017
Last Updated: 23 nov 2017

Year: 2017

ArXiv: 1711.07256 PDF

Abstract:

We consider the Cauchy problem for a gradient flow (GF) generated by a continuously differentiable function in a Hilbert space H and study the reverse approximation of its solutions by the De Giorgi Minimizing Movement approach.

We prove that if H has finite dimension and the driving potential is quadratically bounded from below (in particular if it is Lipschitz) then for every solution u of (GF) (which may have an infinite number of solutions) there exist perturbations depending on the time step and converging to the potential in the Lipschitz norm such that u can be approximated by the perturbed Minimizing Movement scheme.

This result solves a question raised by E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions, eds., Masson, 1993, pp. 81–98.

We also show that even if H has infinite dimension the above approximation holds for the distinguished class of minimal solutions, that generate all the other solutions to the gradient flow by time reparametrization.

Keywords: Gradient flows, minimizing movements, Nonuniqueness, Reverse approximation

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