$C_0$-semigroups generated by second order differential operators with general Wentzell boundary conditions
HTML articles powered by AMS MathViewer
- by Angelo Favini, Giséle Ruiz Goldstein, Jerome A. Goldstein and Silvia Romanelli PDF
- Proc. Amer. Math. Soc. 128 (2000), 1981-1989 Request permission
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
- Ph. Clément and C. A. Timmermans, On $C_0$-semigroups generated by differential operators satisfying Ventcel’s boundary conditions, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 4, 379–387. MR 869754
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Gisèle Ruiz Goldstein, Nonlinear singular diffusion with nonlinear boundary conditions, Math. Methods Appl. Sci. 16 (1993), no. 11, 779–798. MR 1245629, DOI 10.1002/mma.1670161103
- Jerome A. Goldstein and Chin Yuan Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Differential Equations 68 (1987), no. 3, 429–443. MR 891337, DOI 10.1016/0022-0396(87)90179-3
- Jerome A. Goldstein and Chin Yuan Lin, Highly degenerate parabolic boundary value problems, Differential Integral Equations 2 (1989), no. 2, 216–227. MR 984189
- Jerome A. Goldstein and Chin Yuan Lin, Degenerate parabolic problems and the Wentzel boundary condition, Semigroup theory and applications (Trieste, 1987) Lecture Notes in Pure and Appl. Math., vol. 116, Dekker, New York, 1989, pp. 189–199. MR 1009396
- Günter Hellwig, Differentialoperatoren der mathematischen Physik. Eine Einführung, Springer-Verlag, Berlin, 1964 (German). MR 0165398
- Kazuaki Taira, Diffusion processes and partial differential equations, Academic Press, Inc., Boston, MA, 1988. MR 954835
- A.D. Wentzell, On boundary conditions for multidimensional diffusion processes, Theory Prob. and its Appl. 4 (1959), 164-177.
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
- MR Author ID: 333750
- Email: gisele@ceri.memphis.edu
- Jerome A. Goldstein
- Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
- MR Author ID: 74805
- Email: goldstej@msci.memphis.edu
- Silvia Romanelli
- Affiliation: Dipartimento di Matematica, Universita’ di Bari, via E.Orabona, 4 70125 Bari, Italy
- MR Author ID: 237923
- Email: romans@pascal.dm.uniba.it
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1695147