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2 results

Exponential convergence for a convexifyingequation
Guillaume Carlier, Alfred Galichon
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 18 / Issue 3 / July 2012

Published online by Cambridge University Press:

22 July 2011, pp. 611-620

Print publication:

July 2012
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We consider an evolution equation similar to that introduced by Vese in [Comm.Partial Diff. Eq. 24 (1999) 1573–1591] and whose solutionconverges in large time to the convex envelope of the initial datum. We give a stochasticcontrol representation for the solution from which we deduce, under quite generalassumptions that the convergence in the Lipschitz norm is in fact exponential in time.

Generalized Characterizationof the Convex Envelope of a Function
Fethi Kadhi
Journal:

RAIRO - Operations Research / Volume 36 / Issue 1 / January 2002

Published online by Cambridge University Press:

15 July 2002, pp. 95-100

Print publication:

January 2002
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We investigate the minima of functionals of the form $$\int_{[a,b]}g(\dot u(s)){\rm d}s$$ where g is strictly convex. The admissible functions $u:[a,b]\longrightarrow\mathbb{R}$ are not necessarily convex and satisfy $u\leq f$ on [a,b], u(a)=f(a), u(b)=f(b), f is a fixed function on [a,b].We show that the minimum is attained by $\bar f$ , the convex envelope of f.