Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

Sunhi Choi, Inwon C. Kim

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary. Our result is the nonlinear version of the classical result in [3] for divergence-form operators with co-normal boundary data. The main ingredients in our analysis are the estimate on the oscillation on the solutions in half-spaces (Theorem 3.1), and the estimate on the mode of convergence of the solutions as the normal of the half-space varies over irrational directions (Theorem 4.1).

Original languageEnglish (US)
Pages (from-to)419-448
Number of pages30
JournalJournal des Mathematiques Pures et Appliquees
Volume102
Issue number2
DOIs
StatePublished - 2014

Fingerprint

Nonlinear PDE
Homogenization
Half-space
Operator
Oscillation
Rotation Invariant
Fully Nonlinear
Neumann Problem
Boundary Problem
Theorem
Estimate
Averaging
Bounded Domain
Divergence
Vary
Converge

Keywords

  • Comparison principle
  • Homogenization
  • Local barriers
  • Oscillatory neumann conditions
  • Viscosity solutions

ASJC Scopus subject areas

  • Applied Mathematics
  • Mathematics(all)

Cite this

Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data. / Choi, Sunhi; Kim, Inwon C.

In: Journal des Mathematiques Pures et Appliquees, Vol. 102, No. 2, 2014, p. 419-448.

Research output: Contribution to journalArticle

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