Abstract
We use the newly developed Lord Kelvin’s method of images (Bobrowski in J Evol Equ 10(3):663–675, 2010; Semigroup Forum 81(3):435–445, 2010) to show existence of a unique cosine family generated by a restriction of the Laplace operator in C[0, 1] that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.
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The authors are grateful to the anonymous referee for comments improving the paper and in particular for suggesting a simplified proof of Lemma 4.1. The second author is partially supported by the Land Baden–Württemberg in the framework of the Juniorprofessorenprogramm—research project on “Symmetry methods in quantum graphs.”
Adam Bobrowski is on leave from Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland.
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Bobrowski, A., Mugnolo, D. On moments-preserving cosine families and semigroups in C[0, 1]. J. Evol. Equ. 13, 715–735 (2013). https://doi.org/10.1007/s00028-013-0199-x
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DOI: https://doi.org/10.1007/s00028-013-0199-x