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July 2016, 36(7): 3677-3703. doi: 10.3934/dcds.2016.36.3677
Neumann homogenization via integro-differential operators
Nestor Guillen 1, and Russell W. Schwab 2,
Department of Mathematics, University of Massachusetts, Amherst, Amherst, MA 90095, United States
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, United States
Received July 2015 Revised January 2016 Published March 2016
In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is motivated by a new integro-differential representation for nonlinear operators with a comparison principle which we also introduce. In the simple case that the original domain is an infinite strip with almost periodic Neumann data, this leads to an almost periodic homogenization problem involving a fully nonlinear integro-differential operator on the Neumann boundary. This method gives a new proof-- which was left as an open question in the earlier work of Barles- Da Lio- Lions- Souganidis (2008)-- of the result obtained recently by Choi-Kim-Lee (2013), and we anticipate that it will generalize to other contexts.
Keywords: integro-differential representation, nonlocal boundary operators., homogenization, Dirichlet to Neumann.
Mathematics Subject Classification: 35B27, 35J60, 35J99, 35R09, 45K05, 47G20, 49L25, 49N70, 60J75, 93E2.
Citation: Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
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Nestor Guillen Russell W. Schwab