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Perturbing the boundary conditions of the generator of a cosine family

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Abstract

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L:D(A)→∂X and Φ:D(A)→∂X (where ∂X is another Banach space called boundary space). Putting conditions on L and Φ, we show that the second order abstract Cauchy problem for the operator A Φ with A Φ u=Au and domain D(A Φ):={u∈D(A):Lu=Φu} is well-posed and thus it generates a cosine operator function on the Banach space X.

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References

  1. Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  2. Bátkai, A., Engel, K.-J.: Abstract wave equations with generalized Wentzell boundary conditions. J. Differ. Equ. 207, 1–20 (2004)

    Article  MATH  Google Scholar 

  3. Chill, R., Keyantuo, V., Warma, M.: Generation a cosine families on L p(0,1) by elliptic operators with Robin boundary conditions. In: Functional Analysis and Evolutions Equations, Günter Lumer Volume, pp. 113–130 (2007)

    Google Scholar 

  4. Goldstein, J.: Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985)

    MATH  Google Scholar 

  5. Greiner, G.: Perturbing the boundary conditions of a generator. Houst. J. Math. 13, 213–229 (1987)

    MathSciNet  MATH  Google Scholar 

  6. Greiner, G., Kuhn, K.: Linear and semilinear boundary conditions: the analytic case. In: Lecture Notes in Pure and Applied Mathematics, vol. 135, pp. 193–211 (1991)

    Google Scholar 

  7. Hoppe, R.: Interpolation of cosine operator functions. Ann. Mat. Pura Appl. 136, 183–212 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Keyantuo, V., Warma, M.: The wave equation on L p spaces. Semigroup Forum 71, 73–92 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Keyantuo, V., Warma, M.: The wave equation with Wentzell-Robin boundary conditions on L p spaces. J. Differ. Equ. 229, 680–697 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kisyński, J.: On cosine operator functions and one-parameter groups operators. Stud. Math. 44, 93–105 (1969)

    Google Scholar 

  11. Mugnolo, D.: Operator matrices as generators of cosine operator functions. Integral Equ. Oper. Theory 54, 441–464 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nickel, G.: A new look at boundary perturbations of generators. Electron. J. Differ. Equ. 95, 1–14 (2004)

    MathSciNet  Google Scholar 

  13. Piskarev, S., Shaw, S.-Y.: Perturbation and comparison of cosine operator functions. Semigroup Forum 51, 225–246 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Xiao, T.-J., Liang, J.: Second order differential operators with Feller-Wentzell type boundary conditions. J. Funct. Anal. 254, 1467–1486 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Facultad de Ciencias Básicas, Universidad Tecnológica de Bolivar, Cartagena, Colombia

    Edgardo Alvarez-Pardo

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  1. Edgardo Alvarez-Pardo
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Correspondence to Edgardo Alvarez-Pardo.

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Communicated by Jerome A. Goldstein.

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Alvarez-Pardo, E. Perturbing the boundary conditions of the generator of a cosine family. Semigroup Forum 85, 58–74 (2012). https://doi.org/10.1007/s00233-011-9361-3

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  • DOI: https://doi.org/10.1007/s00233-011-9361-3

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