### Abstract

Original language | English |
---|---|

Pages (from-to) | 271-294 |

Number of pages | 24 |

Journal | Asymptotic Analysis |

Volume | 82 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Jan 2013 |

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### Keywords

- homogenization
- evolutionary differential equations
- delay and memory effects
- fractional derivatives
- thermoelasticity

### Cite this

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*Asymptotic Analysis*, vol. 82, no. 3-4, pp. 271-294. https://doi.org/10.3233/ASY-2012-1145

**Homogenization of a class of linear partial differential equations.** / Waurick, Marcus.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Homogenization of a class of linear partial differential equations.

AU - Waurick, Marcus

PY - 2013/1/1

Y1 - 2013/1/1

N2 - We present a unified Hilbert space perspective to homogenization of a class of evolutionary equations of mathematical physics. We formulate homogenization in a purely operator-theoretic setting. Using “A structural observation for linear material laws in classical mathematical physics” by R. Picard [Mathematical Methods in the Applied Sciences 32 (2009), 1768–1803], we discuss constitutive relations as certain elements of the Hardy space ℋ∞(E;L(H)) of bounded, analytic and operator-valued functions M:E→L(H), where E\subseteqq C open, H Hilbert space. The core idea is to introduce a certain topology on the set of constitutive relations. Given a convergent sequence of constitutive relations, the behavior of solutions to the respective problems is discussed. We apply the results to the equations of acoustics, thermodynamics, elasticity or coupled systems such as thermo-elasticity. The respective equations may also incorporate memory or delay terms and fractional derivatives. In particular, constitutive relations via differential equations can also be treated.

AB - We present a unified Hilbert space perspective to homogenization of a class of evolutionary equations of mathematical physics. We formulate homogenization in a purely operator-theoretic setting. Using “A structural observation for linear material laws in classical mathematical physics” by R. Picard [Mathematical Methods in the Applied Sciences 32 (2009), 1768–1803], we discuss constitutive relations as certain elements of the Hardy space ℋ∞(E;L(H)) of bounded, analytic and operator-valued functions M:E→L(H), where E\subseteqq C open, H Hilbert space. The core idea is to introduce a certain topology on the set of constitutive relations. Given a convergent sequence of constitutive relations, the behavior of solutions to the respective problems is discussed. We apply the results to the equations of acoustics, thermodynamics, elasticity or coupled systems such as thermo-elasticity. The respective equations may also incorporate memory or delay terms and fractional derivatives. In particular, constitutive relations via differential equations can also be treated.

KW - homogenization

KW - evolutionary differential equations

KW - delay and memory effects

KW - fractional derivatives

KW - thermoelasticity

U2 - 10.3233/ASY-2012-1145

DO - 10.3233/ASY-2012-1145

M3 - Article

VL - 82

SP - 271

EP - 294

JO - Asymptotic Analysis

JF - Asymptotic Analysis

SN - 0921-7134

IS - 3-4

ER -