Stochastic homogenization of L ∞ variational problems

Scott N. Armstrong; Panagiotis E. Souganidis

doi:10.1016/j.aim.2012.02.018

Stochastic homogenization of L ^∞ variational problems

Scott N. Armstrong, Panagiotis E. Souganidis

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We present a homogenization result for L ^∞ variational problems in general stationary ergodic random environments. By introducing a generalized notion of distance function (a special solution of an associated eikonal equation) and demonstrating a connection to absolute minimizers of the variational problem, we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments.

Original language	English (US)
Pages (from-to)	3508-3535
Number of pages	28
Journal	Advances in Mathematics
Volume	229
Issue number	6
DOIs	https://doi.org/10.1016/j.aim.2012.02.018
State	Published - Apr 1 2012

Keywords

Eikonal equation
Hamilton-Jacobi equation
L calculus of variations
Stochastic homogenization

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2012.02.018

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title = "Stochastic homogenization of L ∞ variational problems",

abstract = "We present a homogenization result for L ∞ variational problems in general stationary ergodic random environments. By introducing a generalized notion of distance function (a special solution of an associated eikonal equation) and demonstrating a connection to absolute minimizers of the variational problem, we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments.",

keywords = "Eikonal equation, Hamilton-Jacobi equation, L calculus of variations, Stochastic homogenization",

author = "Armstrong, {Scott N.} and Souganidis, {Panagiotis E.}",

note = "Funding Information: The first author was partially supported by NSF Grant DMS-1004645 and the second author by NSF Grant DMS-0901802. The first author also thanks Charlie Smart and Vesa Julin for helpful conversations. We acknowledge an anonymous referee for suggesting the example at the end of Section 5.",

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T1 - Stochastic homogenization of L ∞ variational problems

AU - Armstrong, Scott N.

AU - Souganidis, Panagiotis E.

N1 - Funding Information: The first author was partially supported by NSF Grant DMS-1004645 and the second author by NSF Grant DMS-0901802. The first author also thanks Charlie Smart and Vesa Julin for helpful conversations. We acknowledge an anonymous referee for suggesting the example at the end of Section 5.

PY - 2012/4/1

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N2 - We present a homogenization result for L ∞ variational problems in general stationary ergodic random environments. By introducing a generalized notion of distance function (a special solution of an associated eikonal equation) and demonstrating a connection to absolute minimizers of the variational problem, we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments.

AB - We present a homogenization result for L ∞ variational problems in general stationary ergodic random environments. By introducing a generalized notion of distance function (a special solution of an associated eikonal equation) and demonstrating a connection to absolute minimizers of the variational problem, we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments.

KW - Eikonal equation

KW - Hamilton-Jacobi equation

KW - L calculus of variations

KW - Stochastic homogenization

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