On the regularity of abstract Cauchy problems and boundary value problems

Clément, Philippe ; Guerre-Delabrière, Sylvie

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 4, pp. 245-266.

Abstract (VO)

Maximal regularity (in \( L_{p} \)-sense) for abstract Cauchy problems of order one and boundary value problems of order two is studied. In general, regularity of the first problems implies regularity of the second ones; the converse is shown to hold if the underlying Banach space has the UMD property. A stronger notion of regularity, introduced by Sobolevskii, plays an important role in the proofs.

Riassunto

Viene studiata la regolarità massimale (in \( L_{p} \)) di problemi di Cauchy astratti di ordine uno e di problemi al contorno di ordine due. In generale, la regolarità del primo tipo di problemi implica la regolarità del secondo; l’inverso viene dimostrato quando lo spazio di Banach in oggetto ha la proprietà UMD. Una nozione di regolarità forte, introdotta da Sobolevskii, ha un ruolo importante nelle dimostrazioni.

Bibliographie

[1] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Math., 21, 1983, 163-168. | DOI | MR | Zbl

[2] S. Bu - Ph. Clément - S. Guerre-Delabrière, Regularity of pairs of positive operators. Ill. J. of Maths., 1998, to appear. | fulltext mini-dml | MR | Zbl

[3] R. Burkholder, A geometrical characterization of Banach spaces in which martingale differences are unconditional. Ann. Prob., 9, 1981, 997-1011. | fulltext mini-dml | MR | Zbl

[4] P. Butzer - H. Berens, Semi-groups of operators and approximation. Springer-Verlag, NY Inc., 1967. | MR | Zbl

[5] P. Cannarsa - V. Vespri, On maximal \( L_{p} \)-regularity for the abstract Cauchy Problems. Boll. U.M.I., 1986, 165-175. | MR | Zbl

[6] Ph. Clément - P. Grisvard, Somme d’opérateurs et régularité \( L_{p} \) dans les problèmes aux limites. CRAS Paris, t. 314, 1992, 821-824. | MR | Zbl

[7] Ph. Clément - S. Guerre-Delabrière, On the regularity of abstract Cauchy problems of order one and boundary value problems of order two. Publ. Math. Univ. P. et M. Curie (Paris 6), 119, 1997, 1-44. | Zbl

[8] T. Coulhon - D. Lamberton, Régularité \( L_{p} \) pour les equations d’évolution. Séminaire d’analyse fonctionnelle Paris VI-VII, 1984-85, 155-165. | MR | Zbl

[9] G. Da Prato - P. Grisvard, Sommes d’opérateurs linéaires et equations différentielles opérationnelles. J. Math. pures et appl., 54, 1975, 305-387. | MR | Zbl

[10] G. Da Prato - P. Grisvard, Equations d’évolution abstraites non linéaires de type parabolique. Ann. Math. Pura Appl., 120, 1979, 329-396. | DOI | MR | Zbl

[11] G. Da Prato - E. Sinestrari, Differential operators with non dense domain. Ann. Scuola Norm. Sup. Pisa, cl. Sci., 4, 14, 1987, 185-344. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[12] L. De Simon, Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rend. Sem. Mat. Univ. Padova, 1964, 205-223. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[13] G. Di Blasio, Linear parabolic evolution equation in \( L_{p} \)-spaces. Annali Mat., 138, 1984, 55-104. | DOI | MR | Zbl

[14] G. Dore, \( L_{p} \)-regularity of abstract differential equations. Lecture notes in math., 1540, Springer-Verlag, 1993, 25-38. | DOI | MR | Zbl

[15] G. Dore - A. Venni, On the closedness of the sum of two closed operators. Math. Zeitschr., 196, 1987, 189-201. | fulltext EuDML | DOI | MR | Zbl

[16] Y. Giga - H. Sohr, Abstract \( L_{p} \) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. Hokkaido University Preprint, series in Mathematics, series 60, Nov. 1989, 38 pp. | DOI | MR | Zbl

[17] P. Grisvard, Equations différentielles abstraites. Ann. Sc. Ecole Normale Sup., 2, 1969, 311-395. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[18] S. Guerre-Delabrière, \( L_{p} \)-regularity of the Cauchy problem and the geometry of Banach spaces. Ill. J. of Maths., 39, 1995, 556-566. | fulltext mini-dml | MR | Zbl

[19] M. Hieber - J. Prüss, Heat kernel and maximal \( L^{p} \)-\( L^{q} \) estimates for parabolic evolution equations. Comm. PDE, 22, 1997, 1647-1669. | DOI | MR | Zbl

[20] H. Komatsu, Fractional powers of operators. Pacific J. of Math., 19, 1966, 285-346. | fulltext mini-dml | MR | Zbl

[21] C. Le Merdy, Counterexamples on \( L_{p} \)-Maximal regularity. Preprint

[22] R. Nagel (Ed.), One-parameter semi groups of positive operators. Lecture notes in math., 1184, Springer-Verlag, Berlin-New York 1986. | Zbl

[23] A. Pazy, Semi groups of linear operators and applications to partial differential equations. Springer-Verlag, New York 1983. | Zbl

[24] J. Prüss - H. Sohr, On operators with bounded imaginary powers in Banach spaces. Math. Z., 203, 1990, 429-452. | fulltext EuDML | DOI | MR | Zbl

[25] P. E. Sobolevskii, Fractional powers of coercively positive sums of operators. Soviet Math. Dokl., 16, 1975. | MR | Zbl

[26] P. E. Sobolevskii, Coerciveness inequalities for abstract parabolic equations. Dokl. Acad. Nauk, 157, 1964, 52-55. | MR | Zbl

[27] H. Tanabe, Equations of Evolution. Pitman, London, San Francisco, Melbourne 1979. | MR | Zbl

[28] H. Triebel, Interpolation theory, function spaces, differential operators. North Holland, 1978. | MR | Zbl

[29] W. Von Wahl, The equation \( u^{\prime} + Au = f \) in a Hilbert space and \( L^{p} \)-estimates for parabolic equations. J. London Math. Soc., 25, 1982, 483-497. | DOI | MR | Zbl

[30] K. Yosida, Functional Analysis. Springer-Verlag, New York 1971. | Zbl

[31] F. Zimmermann, On vector-valued Fourier multiplier theorems. Studia Math., 93, 1989, 201-222. | fulltext EuDML | fulltext mini-dml | MR | Zbl