### Abstract

Let {Mathematical expression} where a is a smooth periodic matrix and L_{0} is the homogenized operator corresponding to the family (L_{ε}). Let D be a nice domain, and let P_{ε}(x, y), P_{0}(x, y) be the Poisson kernels associated with L_{ε} and L_{0}. We show that in general P_{ε}(x, ·) does not converge strongly to P_{0}(x, ·) in L^{p}, by exhibiting two counterexamples. This result has the following implication in the theory of boundary control of distributed systems: if, with z given, u_{ε}(x) = ∫ P_{ε}(x, y)g(y) and u_{0}(x) = ∫P_{0}(x,y)g(y), then, in general,.

Original language | English (US) |
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Pages (from-to) | 109-119 |

Number of pages | 11 |

Journal | Applied Mathematics & Optimization |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1987 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Mathematics(all)

### Cite this

**Counterexamples related to high-frequency oscillation of Poisson's kernel.** / Avellaneda, Marco; Lin, Fang-Hua.

Research output: Contribution to journal › Article

*Applied Mathematics & Optimization*, vol. 15, no. 1, pp. 109-119. https://doi.org/10.1007/BF01442649

}

TY - JOUR

T1 - Counterexamples related to high-frequency oscillation of Poisson's kernel

AU - Avellaneda, Marco

AU - Lin, Fang-Hua

PY - 1987/1

Y1 - 1987/1

N2 - 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,.

AB - 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,.

UR - http://www.scopus.com/inward/record.url?scp=34250106154&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250106154&partnerID=8YFLogxK

U2 - 10.1007/BF01442649

DO - 10.1007/BF01442649

M3 - Article

VL - 15

SP - 109

EP - 119

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

SN - 0095-4616

IS - 1

ER -