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November  2016, 15(6): 2475-2487. doi: 10.3934/cpaa.2016045

Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains

1. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna

2. 

The University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152

3. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

4. 

Dipartimento di Matematica, Università a di Bologna, 40126 Bologna, Italy

5. 

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

Received  May 2016 Revised  July 2016 Published  September 2016

We study nonsymmetric second order elliptic operators with Wentzell boundary conditions in general domains with sufficiently smooth boundary. The ambient space is a space of $L^p$- type, $1\le p\le \infty$. We prove the existence of analytic quasicontractive $(C_0)$-semigroups generated by the closures of such operators, for any $1< p< \infty$. Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coefficients of the boundary condition. We also specify precisely the domains of the generators explicitly in the case of bounded domains and $1 < p < \infty$, when all the ingredients of the problem, including the boundary of the domain, the coefficients, and the initial condition, are of class $C^{\infty}$.
Citation: 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
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,, \emph{Comm. Appl. Anal.}, 15 (2011), 313.   Google Scholar

[2]

R. P. Clendenen, G. R. Goldstein and J. A. Goldstein, Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations,, \emph{Discrete Continuous Dynam. Systems, 9 (2016), 651.   Google Scholar

[3]

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 \emph{Advances in Nonlinear Analysis: Theory, (2009), 279.   Google Scholar

[4]

G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian,, \emph{Semigroup Forum}, 77 (2008), 101.   Google Scholar

[5]

G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli, Continuous dependence in hyperbolic problems with Wentzell boundary conditions,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 419.   Google Scholar

[6]

K.-J. Engel and G. Fragnelli, Analyticity of semigroups generated by operators with generalized Wentzell boundary conditions,, \emph{Adv. Differential Equations}, 10 (2005), 1301.   Google Scholar

[7]

H. O. Fattorini, The Cauchy Problem,, Addison-Wesley, (1983).   Google Scholar

[8]

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,, \emph{Math. Nachr.}, 283 (2010), 504.   Google Scholar

[9]

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,, \emph{Proc. Amer. Math. Soc.}, 128 (2000), 1981.   Google Scholar

[10]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition,, \emph{J. Evol. Equ.}, 2 (2002), 1.   Google Scholar

[11]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Wentzell boundary conditions in the nonsymmetric case,, \emph{Math. Model. Nat. Phenom.}, 3 (2008), 143.   Google Scholar

[12]

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

[13]

G. R. Goldstein, J. A. Goldstein and M. Pierre, The Agmon-Douglis-Nirenberg problem in the context of dynamic boundary conditions,, in preparation., ().   Google Scholar

[14]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford University Press, (1985).   Google Scholar

[15]

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

[16]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions,, \emph{Electr. J. Diff. Eq.}, 118 (2006), 1.   Google Scholar

[17]

H. Triebel, Theory of Function Spaces,, Birkh\, (1983).   Google Scholar

[18]

H. Vogt and J. Voigt, Wentzell boundary conditions in the context of Dirichlet forms,, \emph{Adv. Differential Equations}, 8 (2003), 821.   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,, \emph{Comm. Appl. Anal.}, 15 (2011), 313.   Google Scholar

[2]

R. P. Clendenen, G. R. Goldstein and J. A. Goldstein, Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations,, \emph{Discrete Continuous Dynam. Systems, 9 (2016), 651.   Google Scholar

[3]

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 \emph{Advances in Nonlinear Analysis: Theory, (2009), 279.   Google Scholar

[4]

G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian,, \emph{Semigroup Forum}, 77 (2008), 101.   Google Scholar

[5]

G. M. Coclite, A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli, Continuous dependence in hyperbolic problems with Wentzell boundary conditions,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 419.   Google Scholar

[6]

K.-J. Engel and G. Fragnelli, Analyticity of semigroups generated by operators with generalized Wentzell boundary conditions,, \emph{Adv. Differential Equations}, 10 (2005), 1301.   Google Scholar

[7]

H. O. Fattorini, The Cauchy Problem,, Addison-Wesley, (1983).   Google Scholar

[8]

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,, \emph{Math. Nachr.}, 283 (2010), 504.   Google Scholar

[9]

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,, \emph{Proc. Amer. Math. Soc.}, 128 (2000), 1981.   Google Scholar

[10]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition,, \emph{J. Evol. Equ.}, 2 (2002), 1.   Google Scholar

[11]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Wentzell boundary conditions in the nonsymmetric case,, \emph{Math. Model. Nat. Phenom.}, 3 (2008), 143.   Google Scholar

[12]

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

[13]

G. R. Goldstein, J. A. Goldstein and M. Pierre, The Agmon-Douglis-Nirenberg problem in the context of dynamic boundary conditions,, in preparation., ().   Google Scholar

[14]

J. A. Goldstein, Semigroups of Linear Operators and Applications,, Oxford University Press, (1985).   Google Scholar

[15]

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

[16]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions,, \emph{Electr. J. Diff. Eq.}, 118 (2006), 1.   Google Scholar

[17]

H. Triebel, Theory of Function Spaces,, Birkh\, (1983).   Google Scholar

[18]

H. Vogt and J. Voigt, Wentzell boundary conditions in the context of Dirichlet forms,, \emph{Adv. Differential Equations}, 8 (2003), 821.   Google Scholar

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