American Institute of Mathematical Sciences

Advanced
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 - A, 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.   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, 3 (2009), 277.   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.   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.   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.   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.   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.   Google Scholar

[9]

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

[10]

J. A. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford Mathematical Monographs, (1985).   Google Scholar

[11]

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

[12]

P. D. Lax, "Functional Analysis,", Wiley-Interscience, (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.   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, 3 (2009), 277.   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.   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.   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.   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.   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.   Google Scholar

[9]

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

[10]

J. A. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford Mathematical Monographs, (1985).   Google Scholar

[11]

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

[12]

P. D. Lax, "Functional Analysis,", Wiley-Interscience, (2002).   Google Scholar

[1]

Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419
[2]

Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020239
[3]

Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401
[4]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047
[5]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023
[6]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Enrico Obrecht, Silvia Romanelli. Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2475-2487. doi: 10.3934/cpaa.2016045
[7]

Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
[8]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[9]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[10]

Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
[11]

Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
[12]

Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144
[13]

Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041
[14]

Mustapha Mokhtar-Kharroubi. Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020244
[15]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[16]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[17]

Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000
[18]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[19]

Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831
[20]

Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences