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Goldstein-Wentzell boundary conditions: Recent results with Jerry and Gisèle Goldstein

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  • We present a survey of recent results concerning heat and telegraph equations, equipped with Goldstein-Wentzell boundary conditions (already known as general Wentzell boundary conditions). We focus on the generation of analytic semigroups and continuous dependence of the solutions of the associated Cauchy problems from the boundary conditions.
    Mathematics Subject Classification: Primary: 46D06, 35K15, 35B30; Secondary: 35L05.

    Citation:

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  • [1]

    T. Clarke, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The Wentzell telegraph equation: Asymptotics and continuous dependence on the boundary conditions, Comm. Appl. Anal., 15 (2011), 313-324.

    [2]

    G. M. Coclite, A. Favini, C. G. Gal, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, The role of Wentzell boundary conditions in linear and nonlinear analysis, in "Advances in Nonlinear Analysis: Theory, Methods and Applications" (ed. S. Sivasundaran), Math. Probl. Eng. Aerosp. Sci., 3, Cambridge Scientific Publishers, Cambridge, (2009), 277-289.

    [3]

    G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian, Semigroup Forum, 77 (2008), 101-108

    [4]

    G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. RomanelliContinuous dependence in hyperbolic problems with Wentzell boundary conditions, Comm. Pure Appl. Math., to appear.

    [5]

    K.-J. Engel and G. Fragnelli, Analyticity of semigroups generated by operators with generalized Wentzell boundary conditions, Adv. Diff. Eqns., 10 (2005), 1301-1320.

    [6]

    A. Favini, G. R. Goldstein, J. A. Goldstein, E. Obrecht and S. Romanelli, Elliptic operators with general Wentzell boundary conditions, analytic semigroups and the angle concavity theorem, Math. Nachr., 283 (2010), 504-521.

    [7]

    A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, $C_0$-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc., 128 (2000), 1981-1989.

    [8]

    A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.

    [9]

    G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Diff. Eqns., 11 (2006), 457-480.

    [10]

    J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.

    [11]

    J. A. Goldstein, On the convergence and approximation of cosine functions, Aeq. Math., 10 (1974), 201-205.

    [12]

    P. D. Lax, "Functional Analysis," Wiley-Interscience, New York, 2002.

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