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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

Angelo Favini 1, , Gisèle Ruiz Goldstein 2, , Jerome A. Goldstein 3, , Enrico Obrecht 4, and  Silvia Romanelli 5,

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 $C^{\infty}$.
Keywords: Wentzell boundary conditions, Nonsymmetric elliptic operators on general domains, continuous dependence., perturbation of symmetric elliptic operators, analytic semigroups.
Mathematics Subject Classification: Primary: 47D06, 47F05, 35K20; Secondary: 35B6.
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 and 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, Comm. Appl. Anal., 15 (2011), 313-324.

[2]

R. P. Clendenen, G. R. Goldstein and J. A. Goldstein, Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations, Discrete Continuous Dynam. Systems, Series S, 9 (2016), 651-660.

[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 Advances in Nonlinear Analysis: Theory, Methods and Applications (ed. S. Sivasundaran), Cambridge Scientific Publishers Ltd., 2009, 279-292.

[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, Semigroup Forum, 77 (2008), 101-108.

[5]

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. Anal., 13 (2014), 419-433.

[6]

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

[7]

H. O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, 1983.

[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, Math. Nachr., 283 (2010), 504-521.

[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, Proc. Amer. Math. Soc., 128 (2000), 1981-1989.

[10]

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.

[11]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Wentzell boundary conditions in the nonsymmetric case, Math. Model. Nat. Phenom., 3 (2008), 143-147.

[12]

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

[13]

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

[14]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985.

[15]

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

[16]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electr. J. Diff. Eq., 118 (2006), 1-20.

[17]

H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.

[18]

H. Vogt and J. Voigt, Wentzell boundary conditions in the context of Dirichlet forms, Adv. Differential Equations, 8 (2003), 821-842.

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.

[2]

R. P. Clendenen, G. R. Goldstein and J. A. Goldstein, Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations, Discrete Continuous Dynam. Systems, Series S, 9 (2016), 651-660.

[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 Advances in Nonlinear Analysis: Theory, Methods and Applications (ed. S. Sivasundaran), Cambridge Scientific Publishers Ltd., 2009, 279-292.

[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, Semigroup Forum, 77 (2008), 101-108.

[5]

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. Anal., 13 (2014), 419-433.

[6]

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

[7]

H. O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, 1983.

[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, Math. Nachr., 283 (2010), 504-521.

[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, Proc. Amer. Math. Soc., 128 (2000), 1981-1989.

[10]

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.

[11]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Wentzell boundary conditions in the nonsymmetric case, Math. Model. Nat. Phenom., 3 (2008), 143-147.

[12]

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

[13]

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

[14]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Oxford, 1985.

[15]

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

[16]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electr. J. Diff. Eq., 118 (2006), 1-20.

[17]

H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.

[18]

H. Vogt and J. Voigt, Wentzell boundary conditions in the context of Dirichlet forms, Adv. Differential Equations, 8 (2003), 821-842.

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