Abstract
Let a ∈ W 1,∞(0,1), a(x) ≥ α > 0, b, c ∈ L ∞ (0,1) and consider the differential operator A given by Au = au″ + bu′ + cu. Let α j , β j (j = 0, 1) be complex numbers satisfying α j , β j ≠ (0,0) for j = 0, 1. We prove that a realization of A with the boundary conditions
generates a cosine family on L p(0, 1) for every p ∈ [1, ∞]. This result is obtained by an explicit calculation, using simply d’Alembert’s formula, of the solutions in the case of the Laplace operator.
We dedicate this work to Günter Lumer in admiration
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander. Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel, 2001.
A. Bátkai and K.-J. Engel. Abstract wave equations with generalized Wentzell boundary conditions. J. Differential Equations 207 (2004), 1–20.
A. Bátkai, K.-J. Engel and M. Haase. Cosine families generated by second-order differential operators on W 1,1(0, 1) with generalized Wentzell boundary conditions. Appl. Anal. 84 (2005), 867–876.
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer Verlag, Heidelberg, Berlin, New-York, 1999.
H.O. Fattorini. Second Order Linear Differential Equations in Banach Spaces. North-Holland Publishing Co., Amsterdam, 1985.
A. Favini, G.R. Goldstein, J.A. Goldstein and S. Romanelli. The one-dimensional wave equation with Wentzell boundary conditions. Differential equations and control theory (Athens, OH, 2000), 139–145, Lecture Notes in Pure and Appl. Math., 225, Dekker, New York, 2002.
J.A. Goldstein. Semigroups of Linear Operators and Applications. Oxford University Press, New York, 1985.
V. Keyantuo and M. Warma. The wave equation on L p-spaces. Semigroup Forum 71 (2005), 73–92.
V. Keyantuo and M. Warma. The wave equation with Wentzell-Robin boundary conditions on L p-spaces. J. Differential Equations 229 (2006), 680–697.
J. Kisyński. On cosine operator functions and one-parameter groups of operators. Studia Math. 44 (1972), 93–105.
J. Kisyński. Semigroups of operators and some of their applications to partial differential equations. Control Theory and Topics in Functional Analysis 3 (1976), 305–405.
J. Malý and W.P. Ziemer. Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.
D. Mugnolo, Operator matrices as generators of cosine operator functions, Integral Eq. Operator Theory 54 (2006), 441–464.
J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993.
T.-J. Xiao and J. Liang. The Cauchy Problem for Higher-order Abstract Differential Equations. Lecture Notes in Mathematics, vol. 1701, Springer, Berlin, 1998.
T.-J. Xiao and J. Liang. A solution to an open problem for wave equations with generalized Wentzell boundary conditions. Math. Ann. 327 (2003), 351–363.
T.-J. Xiao and J. Liang, Second-order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc. 356 (2004), 4787–4809.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Chill, R., Keyantuo, V., Warma, M. (2007). Generation of Cosine Families on L p(0,1) by Elliptic Operators with Robin Boundary Conditions. In: Amann, H., Arendt, W., Hieber, M., Neubrander, F.M., Nicaise, S., von Below, J. (eds) Functional Analysis and Evolution Equations. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7794-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-7643-7794-6_7
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7793-9
Online ISBN: 978-3-7643-7794-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)