Neumann homogenization via integro-differential operators

Nestor  Guillen; Russell W.  Schwab

doi:10.3934/dcds.2016.36.3677

Article Contents

2016, Volume 36, Issue 7: 3677-3703. Doi: 10.3934/dcds.2016.36.3677

This issue Previous Article Spectral properties of renormalization for area-preserving maps Next Article On spatial entropy of multi-dimensional symbolic dynamical systems

Neumann homogenization via integro-differential operators

Nestor Guillen^1, and
Russell W. Schwab^2,

1.
Department of Mathematics, University of Massachusetts, Amherst, Amherst, MA 90095
2.
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824

Received: July 2015

Revised: January 2016

Published: May 2017

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Abstract

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:

Mathematics Subject Classification: 35B27, 35J60, 35J99, 35R09, 45K05, 47G20, 49L25, 49N70, 60J75, 93E20.

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