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February  2014, 34(2): 749-760. doi: 10.3934/dcds.2014.34.749

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

Silvia Romanelli 1,

1. 

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, via E. Orabona 4, 70125 Bari, Italy

Received  June 2012 Revised  May 2013 Published  August 2013

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.
Keywords: Wentzell boundary conditions, Analytic semigroups, continuous dependence on the boundary conditions..
Mathematics Subject Classification: Primary: 46D06, 35K15, 35B30; Secondary: 35L0.
Citation: Silvia Romanelli. Goldstein-Wentzell boundary conditions: Recent results with Jerry and Gisèle Goldstein. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 749-760. doi: 10.3934/dcds.2014.34.749
References:
[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[4]

G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence in hyperbolic problems with Wentzell boundary conditions,, Comm. Pure Appl. Math., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[4]

G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence in hyperbolic problems with Wentzell boundary conditions,, Comm. Pure Appl. Math., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

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