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June 2010 Stochastic homogenization of Hamilon-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities -- Revisited
Pierre-Louis Lions, Panagiotis E. Souganidis
Commun. Math. Sci. 8(2): 627-637 (June 2010).


In this note we revisit the homogenization theory of Hamilton-Jacobi and “viscous”- Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic envi- ronments. We present a new simple proof for the homogenization in probability. The argument uses some a priori bounds (uniform modulus of continuity) on the solution and the convexity and coer- civity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula of the solution as was the case with all previously known proofs. We also introduce a new formula for the effective Hamiltonian for Hamilton-Jacobi and “viscous” Hamilton-Jacobi equations.


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Pierre-Louis Lions. Panagiotis E. Souganidis. "Stochastic homogenization of Hamilon-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities -- Revisited." Commun. Math. Sci. 8 (2) 627 - 637, June 2010.


Published: June 2010
First available in Project Euclid: 25 May 2010

zbMATH: 1197.35031
MathSciNet: MR2664465

Primary: 35B27 , 35D40

Keywords: Hamilton-Jacobi equations , Stochastic homogenization , viscosity solutions

Rights: Copyright © 2010 International Press of Boston

Vol.8 • No. 2 • June 2010
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