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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
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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

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Oxford University Press, Oxford, 1985. Google Scholar

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Wiley- Interscience, New York, 2002. Google Scholar

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Electr. J. Diff. Eq., 118 (2006), 1-20. Google Scholar

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Birkhäuser Verlag, Basel, 1983. Google Scholar

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show all references

References:
[1]

Comm. Appl. Anal., 15 (2011), 313-324. Google Scholar

[2]

Discrete Continuous Dynam. Systems, Series S, 9 (2016), 651-660. Google Scholar

[3]

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

[4]

Semigroup Forum, 77 (2008), 101-108. Google Scholar

[5]

Comm. Pure Appl. Anal., 13 (2014), 419-433. Google Scholar

[6]

Adv. Differential Equations, 10 (2005), 1301-1320. Google Scholar

[7]

Addison-Wesley, Reading, 1983. Google Scholar

[8]

Math. Nachr., 283 (2010), 504-521. Google Scholar

[9]

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

[10]

J. Evol. Equ., 2 (2002), 1-19. Google Scholar

[11]

Math. Model. Nat. Phenom., 3 (2008), 143-147. Google Scholar

[12]

Adv. Diff. Eqns., 11 (2006), 457-480. 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]

Oxford University Press, Oxford, 1985. Google Scholar

[15]

Wiley- Interscience, New York, 2002. Google Scholar

[16]

Electr. J. Diff. Eq., 118 (2006), 1-20. Google Scholar

[17]

Birkhäuser Verlag, Basel, 1983. Google Scholar

[18]

Adv. Differential Equations, 8 (2003), 821-842. Google Scholar

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