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Abstract Elliptic Problems Depending on a Parameter and Parabolic Problems with Dynamic Boundary Conditions

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New Prospects in Direct, Inverse and Control Problems for Evolution Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 10))

Abstract

We study abstract elliptic problems depending on a complex parameter. Such parameter appears also in the boundary conditions. Next, we consider abstract parabolic systems with dynamic boundary conditions. Applications are given to parameter elliptic boundary value problems and to concrete parabolic problems.

Dedicated to the memory of Alfredo Lorenzi

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References

  1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Aliev, B.A., Yakubov, Y.Y.: Elliptic differential-operator problems with a spectral parameter in both the equation and boundary-operator conditions. Adv. Differ. Equ. 11, 1081–1110 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Aliev, B.A., Yakubov, Y.Y.: Erratum to the paper “Elliptic differential-operator problems with a spectral parameter in both the equation and boundary-operator conditions”. Adv. Differ. Equ. 12, 1079 (2006)

    MathSciNet  Google Scholar 

  4. Amann, H.: Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1. Function Spaces. Jindrich Necas Center for Mathematical Modeling Lecture Notes, vol. 23. Matfyzpress, Prague (2009)

    Google Scholar 

  5. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)

    Google Scholar 

  6. Da Prato, G., Grisvard, P., Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl. 54, 301–387 (1975)

    Google Scholar 

  7. Di Cristo, M., Guidetti, D., A. Lorenzi, A.: Abstract parabolic equations with applications to problems in cylindrical space domains. Adv. Differ. Equ. 15, 1–42 (2010)

    Google Scholar 

  8. Denk, R., Hieber, M., Prüss, J.: R-boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type. Memoirs of the American Mathematical Society, vol. 788. American Mathematical Society, Providence (2003)

    Google Scholar 

  9. Dore, G., Venni, A.: On the closeness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Escher, J.: Quasilinear parabolic systems with dynamical boundary conditions. Commun. Partial Differ. Equ. 18, 1309–1364 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Favini, A., Guidetti, D., Yakubov, Y.: Abstract elliptic and parabolic systems with applications to problems in cylindrical domains. Adv. Differ. Equ. 16, 1139–1196 (2011)

    MATH  MathSciNet  Google Scholar 

  12. Grisvard, P.: Commutativité de deux foncteurs d’interpolation et applications I, II. J. Math. Pures Appl. 45, 207–290 (1966)

    MATH  MathSciNet  Google Scholar 

  13. Grisvard, P.: Equations difféntielles abstraites. Ann. Sc. Ec. Norm. Sup. 2, 311–395 (1969)

    MATH  MathSciNet  Google Scholar 

  14. Grisvard, P.: Spazi di tracce e applicazioni. Rend. Mat. 6, 657–729 (1972)

    MathSciNet  Google Scholar 

  15. Guidetti, D.: Generation of analytic semigroups by elliptic operators with dirichlet boundary conditions in a cylindrical domain. Semigroup Forum 68, 108–136 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hintermann, T.: Evolution equations with dynamic boundary conditions. Proc. R. Soc. Edinb. 113, 43–60 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kunstmann, P.C., Weis, L.: Maximal L p-regularity for parabolic equations, fourier multiplier theorems and H ∞-functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Mathematics, vol. 1855. Springer, Berlin (2004)

    Google Scholar 

  18. Mugnolo, D.: Vector-valued heat equations and networks with coupled dynamic boundary conditions. Adv. Differ. Equ. 15, 1125–1160 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  20. Prüss, J.: Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in L p spaces. Math. Bohem. 127, 311–327 (2002)

    MATH  MathSciNet  Google Scholar 

  21. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter, Berlin (1996)

    Book  MATH  Google Scholar 

  22. Simon, J.: Sobolev, Besov and Nikolskii fractional spaces: imbeddings an comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157, 117–148 (1990)

    MATH  Google Scholar 

  23. Tanabe, H.: Equations of Evolution. Monographs and Studies in Mathematics, 6. Pitman, Boston, Mass.-London (1979)

    Google Scholar 

  24. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978). 2nd revised edition Leipzig, Barth (1995)

    Google Scholar 

  25. Triebel, H.: Theory of Function Spaces. Monographs in Mathematics. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  26. Yakubov, S., Yakubov, S.: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman and Hall, London (2000)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Davide Guidetti

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  1. Davide Guidetti
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Correspondence to Davide Guidetti .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Milano, Italy

    Angelo Favini

  2. Dipartimento di Matematica, Università di Bari, Bari, Bari, Italy

    Genni Fragnelli

  3. Dipartimento di Matematica, Università di Bari, Bari, Bari, Italy

    Rosa Maria Mininni

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Guidetti, D. (2014). Abstract Elliptic Problems Depending on a Parameter and Parabolic Problems with Dynamic Boundary Conditions. In: Favini, A., Fragnelli, G., Mininni, R. (eds) New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Springer INdAM Series, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-11406-4_9

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