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Analytic semigroups generated by ultraweak operators

Published online by Cambridge University Press:  14 November 2011

Vincenzo Vespri
Vincenzo Vespri
Affiliation:
Università di Milano, Dipartimento di Matematica, via Saldini 50, 20133 Milano, Italy
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Synopsis

Let Ω be a regular open subset of RN. We present an improved generation result for nonvariational operators in L1 (Ω). This result is obtained by studying ultraweak operators and by proving generation of analytic semigroups in Lp(Ω)(l<p≦∞) and in . We also characterise interpolation and extrapolation spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 1-2 , 1991 , pp. 87 - 105
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Agmon, S.. On the eigenfunctions and the eigenvalues of general elliptic value problems. Comm. Pure Appl. Math. 15 (1962), 119–147.CrossRefGoogle Scholar
2Amann, H.. Dual semigroups and second order linear elliptic value problems. Israel. J. Math. 45 (1983), 25–254.CrossRefGoogle Scholar
3Amann, H.. Semigroups and nonlinear evolution equations. Linear Algebra Appl. 84 (1986), 3–32.CrossRefGoogle Scholar
4Campanato, S.. Proprietá di Hölderianitá di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa 17 (1963), 175–188.Google Scholar
5Campanato, S.. Sistemi ellittici in forma divergenza. Regolaritá all' interne Quederni Scuola Norm. Sup. Pisa (1980).Google Scholar
6Cannarsa, P., Terreni, B. and Vespri, V.. Analytic semigroups generated by nonvariational elliptic systems of second order under Dirichlet boundary conditions. J. Math. Anal. Appl. 112 (1985), 56–103.CrossRefGoogle Scholar
7Cannarsa, P. and Vespri, V.. Analytic semigroups generated on Hölder spaces by second order elliptic systems under Dirichlet boundary conditions. Ann. Mat. Pura. Appl. 140 (1985), 393–415.CrossRefGoogle Scholar
8DaPrato, G. and Grisvard, P.. Maximal regularity for evolution equations by interpolation and extrapolation. J. Fund. Anal. 58 (1984), 107–124.CrossRefGoogle Scholar
9DiBlasio, G.. Perturbation of second order elliptic operators and semilinear evolution equations. NonLinear Anal. 1 (1977), 293–304.CrossRefGoogle Scholar
10DiBlasio, G.. On a class of semilinear parabolic equations in L 1. Differential Equations in Banach Spaces, Proceedings Conf. Bologna LNM 1223 (1986), 74–91.CrossRefGoogle Scholar
11Giaquinta, M.. Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies (Princeton, N.J: Princeton University Press, 1983).Google Scholar
12Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, 2nd edn, (Berlin: Springer, 1983).Google Scholar
13Kato, T. and Tanabe, H.. On the abstract evolution equations. Osaka Math. J. 14 (1962), 107–133.Google Scholar
14Krein, S. G.. Linear differential equations in Banach spaces, Transl. Math. Monographs (Providence, R.I.: American Mathematical Society, 1971).Google Scholar
15Lions, J. L.. Theorems de trace et d'interpolation I. Ann. Scuola. Norm. Sup. Pisa 13 (1953), 389–403.Google Scholar
16Lions, J. L.. Equations différentielles opérationelles et problémes aux limites. Grundlehren Math. Wis. Bd III (Berlin: Springer, 1961).Google Scholar
17Lions, J. L.. Quelques méthodes de résolution des problèmes aux limites non linéaires (Paris: Dunod, Gauthier-Villers, 1969).Google Scholar
18Lunardi, A..Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations. Math. Nachr. 121 (1985), 295–318.CrossRefGoogle Scholar
19Nečas, J.. Les méthodes directes en théorie des equations elliptiques (Paris: Masson, 1967).Google Scholar
20Park, D. G.. Initial boundary value problem for parabolic equations in L 1. Proc. Japan Acad. 62 (1986), 178–180.Google Scholar
21Park, D. G. and Tanabe, H.. On the asymptotic behaviour of linear parabolic equations in L 1 spaces. Ann Scuola Norm. Sup. Pisa.Google Scholar
22Pazy, A.. Asymptotic behaviour of the solution of an abstract evolution equation and some applications. J. Differential Equations 4 (1968), 495–509.CrossRefGoogle Scholar
23Pazy, A.. Semigroups of linear operators and application to partial differential equations (Berlin: Springer, 1983).CrossRefGoogle Scholar
24Sinestrari, E.. On the abstract Cauchy problem of parabolic type in spaces of continuous functions. J. Math. Anal. Appl. 107 (1985), 16–66.CrossRefGoogle Scholar
25Stewart, H. B.. Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc. 199 (1974), 141–162.CrossRefGoogle Scholar
26Tanabe, H.. Convergence to a stationary state of the solution of some kinds of differential equations in a Banach space. Proc. Japan Acad. 37 (1961), 127–130.Google Scholar
27Tanabe, H.. On semilinear equations of elliptic and parabolic type, functional analysis and numerical analysis. Japan–France seminar 1976, 455–463. (Japan society for the promotion of science, 1978).Google Scholar
28Triebel, H.. Interpolation theory, function spaces, differential operators. (Amsterdam: North Holland, 1978).Google Scholar
29Vespri, V.. The functional space C −1, α and analytic semigroups. Differential and Integral Equations 1 (1988), 473–493.CrossRefGoogle Scholar
30Zaidman, S.. Abstract differential equations, Pitman Adv. Publ. Program, RNM: Pitman, (San Francisco 1979).Google Scholar
31Zaidman, S.. Improving regularity of a weak solution of an abstract differential equation. Ann. Soc. Math. Québec 10 (1986), 103–108.Google Scholar