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 & Continuous Dynamical Systems - B, 2015, 20 (7) : 1877-1895. doi: 10.3934/dcdsb.2015.20.1877
References:
[1]

E. Alvarez-Pardo, Perturbing the boundary conditions of the generator of a cosine family,, Semigroup Forum, 85 (2012), 58.  doi: 10.1007/s00233-011-9361-3.  Google Scholar

[2]

E. Alvarez-Pardo and M. Warma, The one-dimensional wave equation with general boundary conditions,, Archiv der Mathematik, 96 (2011), 177.  doi: 10.1007/s00013-010-0209-y.  Google Scholar

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J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates,, J. Math. Anal. Appl., 391 (2012), 312.  doi: 10.1016/j.jmaa.2012.02.002.  Google Scholar

[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.   Google Scholar

[5]

A. Bobrowski, Generation of cosine families via Lord Kelvin's method of images,, J. Evol. Equ., 10 (2010), 663.  doi: 10.1007/s00028-010-0065-z.  Google Scholar

[6]

A. Bobrowski, Lord Kelvin's method of images in semigroup theory,, Semigroup Forum, 81 (2010), 435.  doi: 10.1007/s00233-010-9230-5.  Google Scholar

[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.  doi: 10.1007/s00233-013-9561-0.  Google Scholar

[8]

A. Bobrowski and D. Mugnolo, On moments-preserving cosine families and semigroups in C[0,1],, J. Evol. Equ., 13 (2013), 715.  doi: 10.1007/s00028-013-0199-x.  Google Scholar

[9]

J. R. Cannon, The solution of the heat equation subject to the specification of energy,, Quart. Appl. Math., 21 (1963), 155.   Google Scholar

[10]

R. E. Edwards, Functional Analysis. Theory and Applications,, Dover Publications, (1995).   Google Scholar

[11]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II,, Second edition, (1971).   Google Scholar

[12]

J. A. Goldstein, On the convergence and approximation of cosine functions,, Aequationes Math., 11 (1974), 201.   Google Scholar

[13]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Mathematical Monographs, (1985).   Google Scholar

[14]

G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213.   Google Scholar

[15]

Y. Konishi, Cosine functions of operators in locally convex spaces,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 18 (): 443.   Google Scholar

[16]

A. C. McBride, A. L. Smith and W. Lamb, Strongly differentiable solutions of the discrete coagulation-fragmentation equation,, Phys. D, 239 (2010), 1436.  doi: 10.1016/j.physd.2009.03.013.  Google Scholar

[17]

D. Mugnolo and S. Nicaise, Diffusion processes on an interval under linear moment conditions,, Semigroup Forum, 88 (2014), 479.  doi: 10.1007/s00233-013-9552-1.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.12.016.  Google Scholar

[19]

H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods,, Blaisdell Publishing Co. Ginn and Co., (1965).   Google Scholar

[20]

T.-J. Xiao and J. Liang, Second order differential operators with Feller-Wentzell type boundary conditions,, J. Funct. Anal., 254 (2008), 1467.  doi: 10.1016/j.jfa.2007.12.012.  Google Scholar

show all references

References:
[1]

E. Alvarez-Pardo, Perturbing the boundary conditions of the generator of a cosine family,, Semigroup Forum, 85 (2012), 58.  doi: 10.1007/s00233-011-9361-3.  Google Scholar

[2]

E. Alvarez-Pardo and M. Warma, The one-dimensional wave equation with general boundary conditions,, Archiv der Mathematik, 96 (2011), 177.  doi: 10.1007/s00013-010-0209-y.  Google Scholar

[3]

J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates,, J. Math. Anal. Appl., 391 (2012), 312.  doi: 10.1016/j.jmaa.2012.02.002.  Google Scholar

[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.   Google Scholar

[5]

A. Bobrowski, Generation of cosine families via Lord Kelvin's method of images,, J. Evol. Equ., 10 (2010), 663.  doi: 10.1007/s00028-010-0065-z.  Google Scholar

[6]

A. Bobrowski, Lord Kelvin's method of images in semigroup theory,, Semigroup Forum, 81 (2010), 435.  doi: 10.1007/s00233-010-9230-5.  Google Scholar

[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.  doi: 10.1007/s00233-013-9561-0.  Google Scholar

[8]

A. Bobrowski and D. Mugnolo, On moments-preserving cosine families and semigroups in C[0,1],, J. Evol. Equ., 13 (2013), 715.  doi: 10.1007/s00028-013-0199-x.  Google Scholar

[9]

J. R. Cannon, The solution of the heat equation subject to the specification of energy,, Quart. Appl. Math., 21 (1963), 155.   Google Scholar

[10]

R. E. Edwards, Functional Analysis. Theory and Applications,, Dover Publications, (1995).   Google Scholar

[11]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II,, Second edition, (1971).   Google Scholar

[12]

J. A. Goldstein, On the convergence and approximation of cosine functions,, Aequationes Math., 11 (1974), 201.   Google Scholar

[13]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford Mathematical Monographs, (1985).   Google Scholar

[14]

G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213.   Google Scholar

[15]

Y. Konishi, Cosine functions of operators in locally convex spaces,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 18 (): 443.   Google Scholar

[16]

A. C. McBride, A. L. Smith and W. Lamb, Strongly differentiable solutions of the discrete coagulation-fragmentation equation,, Phys. D, 239 (2010), 1436.  doi: 10.1016/j.physd.2009.03.013.  Google Scholar

[17]

D. Mugnolo and S. Nicaise, Diffusion processes on an interval under linear moment conditions,, Semigroup Forum, 88 (2014), 479.  doi: 10.1007/s00233-013-9552-1.  Google Scholar

[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.  doi: 10.1016/j.jde.2013.12.016.  Google Scholar

[19]

H. F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods,, Blaisdell Publishing Co. Ginn and Co., (1965).   Google Scholar

[20]

T.-J. Xiao and J. Liang, Second order differential operators with Feller-Wentzell type boundary conditions,, J. Funct. Anal., 254 (2008), 1467.  doi: 10.1016/j.jfa.2007.12.012.  Google Scholar

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