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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$C_{0}$-semigroups generated by second order differential operators with general
Wentzell boundary conditions


Authors: Angelo Favini, Giséle Ruiz Goldstein, Jerome A. Goldstein and Silvia Romanelli
Journal: Proc. Amer. Math. Soc. 128 (2000), 1981-1989
MSC (2000): Primary 47D06, 47H06, 35J25
Published electronically: February 16, 2000
MathSciNet review: 1695147
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Abstract: Let us consider the operator $\widetilde {A}u(x)=\phi (x,u'(x))u''(x),$ where $\phi $ is positive and continuous in $(0,1)\times \mathbf{R}$ and $\widetilde {A}$ is equipped with the so-called generalized Wentzell boundary condition which is of the form $a\widetilde {A} u+bu'+cu=0$ at each boundary point, where $(a,b,c)\neq (0,0,0).$ This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions.

Under suitable assumptions on $\phi $, we prove that $\widetilde {A}$ generates a positive $C_{0}$-semigroup on $C[0,1]$ and, hence, many previous (linear or nonlinear) results are extended substantially.


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Additional Information

Angelo Favini
Affiliation: Dipartimento di Matematica, Universita’ di Bologna, Piazza di Porta S.Donato, 5 40127 Bologna, Italy
Email: favini@dm.unibo.it

Giséle Ruiz Goldstein
Affiliation: CERI, University of Memphis, Memphis, Tennessee 38152
Email: gisele@ceri.memphis.edu

Jerome A. Goldstein
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: goldstej@msci.memphis.edu

Silvia Romanelli
Affiliation: Dipartimento di Matematica, Universita’ di Bari, via E.Orabona, 4 70125 Bari, Italy
Email: romans@pascal.dm.uniba.it

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05486-1
PII: S 0002-9939(00)05486-1
Keywords: $C_{0}$-semigroups on $C[0,1]$, nonlinear second order differential operators, generalized Wentzell boundary condition
Received by editor(s): August 15, 1998
Published electronically: February 16, 2000
Additional Notes: This work was supported by M.U.R.S.T. 60$%$ and 40$%$ and by G.N.A.F.A. of C.N.R
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society