# Exponential convergence for a convexifying equation

Guillaume Carlier; Alfred Galichon

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 3, page 611-620
- ISSN: 1292-8119

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topCarlier, Guillaume, and Galichon, Alfred. "Exponential convergence for a convexifying equation." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 611-620. <http://eudml.org/doc/277828>.

@article{Carlier2012,

abstract = {We consider an evolution equation similar to that introduced by Vese in [Comm.
Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution
converges in large time to the convex envelope of the initial datum. We give a stochastic
control representation for the solution from which we deduce, under quite general
assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
},

author = {Carlier, Guillaume, Galichon, Alfred},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Convex envelope; viscosity solutions; stochastic control representation; nonautonomous gradient flows; convex envelope},

language = {eng},

month = {11},

number = {3},

pages = {611-620},

publisher = {EDP Sciences},

title = {Exponential convergence for a convexifying equation},

url = {http://eudml.org/doc/277828},

volume = {18},

year = {2012},

}

TY - JOUR

AU - Carlier, Guillaume

AU - Galichon, Alfred

TI - Exponential convergence for a convexifying equation

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2012/11//

PB - EDP Sciences

VL - 18

IS - 3

SP - 611

EP - 620

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

LA - eng

KW - Convex envelope; viscosity solutions; stochastic control representation; nonautonomous gradient flows; convex envelope

UR - http://eudml.org/doc/277828

ER -

## References

top- O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints. J. Math. Pures Appl.76 (1997) 265–288. Zbl0890.49013
- L. Caffarelli and X. Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications43. American Mathematical Society, Providence, RI (1995).
- Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geom. 33 (1991) 749–786. Zbl0696.35087
- M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc.27 (1992) 1–67.
- W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, Graduate Studies in Mathematics58. Applications of Mathematics, Springer-Verlag (1993).
- B. Kirchheim and J. Kristensen, Differentiability of convex envelopes. C. R. Acad. Sci. Paris Sér. I Math.333 (2001) 725–728. Zbl1053.49013
- A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem. Proc. Amer. Math. Soc.135 (2007) 1689–1694. Zbl1190.35107
- A. Oberman, Computing the convex envelope using a nonlinear partial differential equation. Math. Mod. Methods Appl. Sci.18 (2008) 759–780. Zbl1154.35056
- A. Oberman and L. Silvestre, The Dirichlet Problem for the Convex Envelope. Trans. Amer. Math. Soc. (to appear). Zbl1244.35056
- H.M. Soner and N. Touzi, Stochastic representation of mean curvature type geometric flows. Ann. Probab.31 (2003) 1145–1165. Zbl1080.60076
- N. Touzi, Stochastic control and application to Finance. Lecture Notes available at . URIhttp://www.cmap.polytechnique.fr/˜touzi/
- L. Vese, A method to convexify functions via curve evolution. Comm. Partial Diff. Eq.24 (1999) 1573–1591. Zbl0935.35087

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