Journal
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
Volume 41, Issue 12, Pages 1839-1859Publisher
TAYLOR & FRANCIS INC
DOI: 10.1080/03605302.2016.1238482
Keywords
Probability measure in Hilbert spaces; random fields; semilinear elliptic equation; stochastic homogenization; variational problem
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Funding
- NSF [DMS-1408867, DMS-1515150]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1515150] Funding Source: National Science Foundation
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We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green's function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.
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