Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-00-05486-1
Published electronically: February 16, 2000
MathSciNet review: 1695147
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Ph. Clément and C. A. Timmermanns, On $C_{0}$-semigroups generated by differential operators satisfying Ventcel's boundary conditions, Indag. Math. 89 (1986), 379 - 387. MR 88c:47075
  • 2. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der mathematischen Wissenschaften, Band 224, Springer-Verlag, Berlin, New York, 1983. MR 86c:35035
  • 3. G.R. Goldstein, Nonlinear singular diffusion with nonlinear boundary conditions, Math. Meth. Appl. Sci. 16 (1993), 779-798. MR 94i:35089
  • 4. J.A. Goldstein and C.-Y. Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Differential Equations 68 (1987), 429-443. MR 89e:35079
  • 5. -, Highly degenerate parabolic boundary value problems, Differential $\&$ Integral Equations 2 (1989), 216-227. MR 90b:35131
  • 6. -, Degenerate parabolic problems and the Wentzell boundary condition, Semigroup Theory and Applications, Ph.Clément, S.Invernizzi et al. (eds), vol. 116, Lect. Notes in Pure and Appl. Math., M. Dekker, New York, 1989, pp. 189-199. MR 90h:35133
  • 7. G. Hellwig, Differentialoperatoren der Mathematischen Physik, Springer-Verlag, Berlin, Heidelberg, 1964. MR 29:2682
  • 8. K.Taira, Diffusion Processes and Partial Differential Equations, Academic Press, San Diego, New York, London, Tokyo, 1988. MR 90m:60089
  • 9. A.D. Wentzell, On boundary conditions for multidimensional diffusion processes, Theory Prob. and its Appl. 4 (1959), 164-177.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47D06, 47H06, 35J25

Retrieve articles in all journals with MSC (2000): 47D06, 47H06, 35J25


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: https://doi.org/10.1090/S0002-9939-00-05486-1
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

American Mathematical Society