Functionals-preserving cosine families generated by Laplace operators in C[0,1]

Adam  Bobrowski; Adam  Gregosiewicz; Małgorzata  Murat

doi:10.3934/dcdsb.2015.20.1877

Article Contents

2015, Volume 20, Issue 7: 1877-1895. Doi: 10.3934/dcdsb.2015.20.1877

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Functionals-preserving cosine families generated by Laplace operators in C[0,1]

1.
Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin

Received: April 30, 2014

Revised: February 28, 2015

Published: August 2015

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Abstract

Let $ C[0,1] $ be the space of continuous functions on the unit interval $ [0,1] $. A cosine family $\{C(t), t \in \mathbb{R}\}$ in $C[0,1]$ is said to be Laplace-operator generated, if its generator is a restriction of the Laplace operator $L\colon f \mapsto f''$ to a suitable subset of $C^2[0,1].$ The family is said to preserve a functional $F \in (C[0,1])^*$ if for all $f \in C[0,1]$ and $t \in \mathbb{R}, $ $FC(t)f = Ff.$ We study a class of pairs of functionals such that for each member of this class there is a unique Laplace-operator generated cosine family that preserves both functionals in the pair.

Keywords:

Method of images,
cosine families,
strongly-continuous semigroups,
differential operators with integral conditions.

Mathematics Subject Classification: 47D06, 47D09.

Citation:

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