Counterexamples related to high-frequency oscillation of Poisson's kernel

Marco Avellaneda, Fang Hua Lin

Research output: Contribution to journalArticlepeer-review


Let {Mathematical expression} where a is a smooth periodic matrix and L0 is the homogenized operator corresponding to the family (Lε). Let D be a nice domain, and let Pε(x, y), P0(x, y) be the Poisson kernels associated with Lε and L0. We show that in general Pε(x, ·) does not converge strongly to P0(x, ·) in Lp, by exhibiting two counterexamples. This result has the following implication in the theory of boundary control of distributed systems: if, with z given, uε(x) = ∫ Pε(x, y)g(y) and u0(x) = ∫P0(x,y)g(y), then, in general,.

Original languageEnglish (US)
Pages (from-to)109-119
Number of pages11
JournalApplied Mathematics & Optimization
Issue number1
StatePublished - Jan 1987

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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