# American Institute of Mathematical Sciences

September  2015, 20(7): 1877-1895. doi: 10.3934/dcdsb.2015.20.1877

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

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

Received  May 2014 Revised  March 2015 Published  July 2015

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.
Citation: Adam Bobrowski, Adam Gregosiewicz, Małgorzata Murat. Functionals-preserving cosine families generated by Laplace operators in C[0,1]. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1877-1895. doi: 10.3934/dcdsb.2015.20.1877
##### References:
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##### References:
 [1] E. Alvarez-Pardo, Perturbing the boundary conditions of the generator of a cosine family, Semigroup Forum, 85 (2012), 58-74. doi: 10.1007/s00233-011-9361-3. [2] E. Alvarez-Pardo and M. Warma, The one-dimensional wave equation with general boundary conditions, Archiv der Mathematik, 96 (2011), 177-186. doi: 10.1007/s00013-010-0209-y. [3] J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, J. Math. Anal. Appl., 391 (2012), 312-322. doi: 10.1016/j.jmaa.2012.02.002. [4] A. Bielecki, Une remarque sur la méthode de Banach-Cacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci. Cl. III., 4 (1956), 261-264. [5] A. Bobrowski, Generation of cosine families via Lord Kelvin's method of images, J. Evol. Equ., 10 (2010), 663-675. doi: 10.1007/s00028-010-0065-z. [6] A. Bobrowski, Lord Kelvin's method of images in semigroup theory, Semigroup Forum, 81 (2010), 435-445. doi: 10.1007/s00233-010-9230-5. [7] A. Bobrowski and A. Gregosiewicz, A general theorem on generation of moments-preserving cosine families by Laplace operators in C[0,1], Semigroup Forum, 88 (2014), 689-701. doi: 10.1007/s00233-013-9561-0. [8] A. Bobrowski and D. Mugnolo, On moments-preserving cosine families and semigroups in C[0,1], J. Evol. Equ., 13 (2013), 715-735. doi: 10.1007/s00028-013-0199-x. [9] J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21 (1963), 155-160. [10] R. E. Edwards, Functional Analysis. Theory and Applications, Dover Publications, Inc., New York, 1995. [11] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. [12] J. A. Goldstein, On the convergence and approximation of cosine functions, Aequationes Math., 11 (1974), 201-205. [13] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. [14] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [15] Y. Konishi, Cosine functions of operators in locally convex spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 18 (1971/72), 443-463. [16] A. C. McBride, A. L. Smith and W. Lamb, Strongly differentiable solutions of the discrete coagulation-fragmentation equation, Phys. D, 239 (2010), 1436-1445. doi: 10.1016/j.physd.2009.03.013. [17] D. Mugnolo and S. Nicaise, Diffusion processes on an interval under linear moment conditions, Semigroup Forum, 88 (2014), 479-511. doi: 10.1007/s00233-013-9552-1. [18] D. Mugnolo and S. Nicaise, Well-posedness and spectral properties of heat and wave equations with non-local conditions, J. Differential Equations, 256 (2014), 2115-2151. doi: 10.1016/j.jde.2013.12.016. [19] H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, Blaisdell Publishing Co. Ginn and Co., New York-Toronto-London, 1965. [20] T.-J. Xiao and J. Liang, Second order differential operators with Feller-Wentzell type boundary conditions, J. Funct. Anal., 254 (2008), 1467-1486. doi: 10.1016/j.jfa.2007.12.012.
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