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Fractional powers of operators corresponding to coercive problems in Lipschitz domains. (English. Russian original) Zbl 1287.47030
Funct. Anal. Appl. 47, No. 2, 83-95 (2013); translation from Funkts. Anal. Prilozh. 47, No. 2, 2-17 (2013).
In this paper, the authors are interested in the study of the domains of fractional powers of positive operators.

MSC:
47B65 Positive linear operators and order-bounded operators
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[1] Sh. Agmon, ”On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems,” Comm. Pure Appl. Math., 15:2 (1962), 119–147. · Zbl 0109.32701 · doi:10.1002/cpa.3160150203
[2] M. S. Agranovich, ”Regularity of variational solutions to linear boundary value problems in Lipschitz domains,” Funkts. Anal. Prilozhen., 40:4 (2006), 83–103; English transl.: Functional Anal. Appl., 40:4 (2006), 313–329. · Zbl 1169.35327 · doi:10.4213/faa849
[3] M. S. Agranovich, ”To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz domains,” Funkts. Anal. Prilozhen., 41:4 (2007), 1–21; English transl.: Functional Anal. Appl., 41:4 (2007), 247–263. · Zbl 1159.35319 · doi:10.4213/faa2875
[4] M. S. Agranovich, ”Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains,” Funkts. Anal. Prilozhen., 43:3 (2009), 3–25; English transl.: Functional Anal. Appl., 43:3 (2009), 165–183. · Zbl 1272.47065 · doi:10.4213/faa2964
[5] M. S. Agranovich, ”Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface,” Funkts. Anal. Prilozhen., 45:1 (2011), 1–15; English transl.: Functional Anal. Appl., 45:1 (2011), 1–12. · Zbl 1271.35023
[6] M. S. Agranovich, ”Mixed problems in a Lipschitz domain for strongly elliptic second-order systems,” Funkts. Anal. Prilozhen., 45:2 (2011), 1–22; English transl.: Functional Anal. Appl., 45:2 (2011), 81–98. · Zbl 1271.35023
[7] M. S. Agranovich, ”Remarks on strongly elliptic second-order systems in Lipschitz domains,” Russian J. Math. Phys., 20:4 (2012), 405–416. · Zbl 1266.35048 · doi:10.1134/S1061920812040012
[8] M. S. Agranovich, Sobolev Spaces, Their Generalizatrions, and Elliptic Problems in Smooth and Lipschitz Domains [in Russian], MTsNMO, Moscow, 2013.
[9] M. S. Agranovich and M. I. Vishik, ”Elliptic problems with a parameter and parabolic problems of general type,” UspekhiMat. Nauk, 19:3(117) (1964), 53–161; English transl.: Russian Math. Surveys, 19:3 (1964), 53–157. · Zbl 0137.29602
[10] W. Arendt, ”Semigroups and evolution equations: functional calculus, regularity and kernel estimates,” in: Handbook of Differential Equations, Evolutionary Differential Equations, vol. 1, Elsevier/North-Holland, Amsterdam, 2004, 1–85. · Zbl 1082.35001
[11] P. Auscher, N. Badr, R. Haller-Dintelmann, and J. Rehberg, The square root problem for second order, divergence form operators with mixed boundary conditions on L p, http://arxiv.org/abs/1210.0780v1 . · Zbl 1333.47034
[12] P. Auscher, S. Hofmann, M. Lacey, J. Lewis, A. McIntosh, and P. Tchamitchian, ”The solution of Kato’s conjectures,” C. R. Acad. Sci. Paris, Sér. 1, 332:7 (2001), 601–606. · Zbl 1017.47034 · doi:10.1016/S0764-4442(01)01893-6
[13] P. Auscher, A. McIntosh, and A. Nahmod, ”Holomorphic functional calculi of operators, quadratic estimates and interpolation,” Indiana Univ. Math. J., 46:2 (1997), 375–403. · Zbl 0903.47011
[14] P. Auscher, S. Hofmann, A. McIntosch, and P. Tchamitchian, ”The Kato square root problem for higher order elliptic operators and systems on \(\mathbb{R}\)n,” J. Evol. Equ., 1:4 (2001), 361–385. · Zbl 1019.35029 · doi:10.1007/PL00001377
[15] P. Auscher and P. Tchamitchian, ”Square root problem for divergence operators and related topics,” Astérisque, 249 (1998), 1–171. · Zbl 0909.35001
[16] P. Auscher and P. Tchamitchian, ”Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L 2 theory,” J. Anal. Math., 90 (2003), 1–12. · Zbl 1173.35420 · doi:10.1007/BF02786549
[17] P. Auscher and P. Tchamitchian, ”Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L p theory,” Math. Ann., 320:3 (2001), 577–623. · Zbl 1161.35350 · doi:10.1007/PL00004487
[18] A. Axelsson, S. Keith, and A. McIntosh, ”The Kato square root problem for mixed boundary value problems,” J. London Math. Soc., 74:1 (2006), 113–130. · Zbl 1123.35013 · doi:10.1112/S0024610706022873
[19] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[20] S. Blunck and P. Kunstmann, ”Calderón-Zygmund theory for non-integral operators and the H functional calculus,” Rev. Mat. Iberoamericana, 19:3 (2003), 919–942. · Zbl 1057.42010 · doi:10.4171/RMI/374
[21] A. F. M. ter Elst and D. W. Robinson, ”On Kato’s square root problem,” Hokkaido Math. J., 26:2 (1997), 365–376. · Zbl 0884.35016
[22] J. Griepentrog, K. Gröger, H.-Ch. Kaiser, and J. Rehberg, ”Interpolation for function spaces related to mixed boundary value problems,” Math. Nachr., 241 (2002), 110–120. · Zbl 1010.46021 · doi:10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R
[23] D. Grisvard, ”Caractérisation de quelques espaces d’interpolation,” Arc. Rational Mech. Anal., 25 (1967), 40–63. · Zbl 0187.05901 · doi:10.1007/BF00281421
[24] M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006. · Zbl 1101.47010
[25] S. Hoffmann, ”A short course on the Kato problem,” Contemp. Math., 289 (2001), 61–67. · doi:10.1090/conm/289/04875
[26] L. Hörmander, Linear Partial Differential Operators, Academic Press, New York, 1963. · Zbl 0108.09301
[27] T. Hytönen, A. McIntosh, and P. Portal, ”Kato’s square root problem in Banach spaces,” J. Funct. Anal., 254:3 (2008), 675–726. · Zbl 1143.47013 · doi:10.1016/j.jfa.2007.10.006
[28] S. Janson, P. Nilsson, and J. Peetre, ”Notes on Wolff’s note on interpolation spaces,” Proc. London Math. Soc. (3), 48:2 (1984), 283–299. · Zbl 0532.46046 · doi:10.1112/plms/s3-48.2.283
[29] T. Kato, ”Fractional powers of dissipative operators,” J. Math. Soc. Japan, 13 (1961), 246–274. · Zbl 0113.10005 · doi:10.2969/jmsj/01330246
[30] T. Kato, ”Fractional powers of dissipative operators, II,” J. Math. Soc. Japan, 14 (1962), 242–248. · Zbl 0108.11203 · doi:10.2969/jmsj/01420242
[31] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1966. · Zbl 0148.12601
[32] H. Komatsu, ”Fractional powers of operators,” Pacif. J. Math., 19 (1966), 285–346. · Zbl 0154.16104 · doi:10.2140/pjm.1966.19.285
[33] M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Integrable Functions [in Russian], Nauka, Moscow, 1966.
[34] J. L. Lions, Équations différentielles operationelles et problèmes aux limites, Springer-Verlag, Berlin etc., 1961. · Zbl 0098.31101
[35] J. L. Lions, ”Espaces d’interpolation et domaines de puissances fractionaires d’opeŕateurs,” J. Math. Soc. Japan, 14 (1962), 233–241. · Zbl 0108.11202 · doi:10.2969/jmsj/01420233
[36] J.-L. Lions and E. Magenes, Problèmes aux limites non homogenes et applications, vol. 1, Dunod, Paris, 1968. · Zbl 0165.10801
[37] A. McIntosh, ”On the compatibility of A 1/2 and A*1/2,” Proc. Amer. Math. Soc., 32:2 (1972), 430–434. · Zbl 0248.47020
[38] A. McIntosh, ”Square roots of elliptic operators,” J. Funct. Anal., 61:3 (1985), 307–327. · Zbl 0592.47043 · doi:10.1016/0022-1236(85)90025-4
[39] A. McIntosh, ”Operators which have an H functional calculus,” in: Miniconference on Operator Theory and Partial Differential Equations, Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, 210–231.
[40] A. McIntosh, The Square Root Problem for Elliptic Operators: a Survey, Lecture Notes in Math., vol. 1450, Springer-Verlag, Berlin, 1990. · Zbl 0723.47032
[41] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge, UK, 2000. · Zbl 0948.35001
[42] J. Nečas, Les méthodes directes en théorie des equations elliptiques, Masson, Paris, 1967; Direct Methods in the Theory of Elliptic Equations, Springer-Verlag, Berlin-Heidelberg, 2012.
[43] L. Nirenberg, ”Remarks on strongly elliptic partial differential equations,” Comm. Pure Appl. Math., 8 (1965), 649–675. · Zbl 0067.07602
[44] R. T. Seeley, ”Norms and domains of the complex powers Az B,” Amer. J. Math., 93:2 (1971), 299–309. · Zbl 0218.35034 · doi:10.2307/2373377
[45] R. T. Seeley, ”Interpolation in Lp with boundary conditions,” Studia Math., 44 (1972), 47–60. · Zbl 0237.46041
[46] A. M. Selitskii, ”The space of initial data of the 3d boundary value problem for a parabolic differential-difference equation in the one-dimensional case,” Mat. Zametki, 92:4 (2012), 636–640; English transl.: Math. Notes, 92:4 (2012), 580–584. · doi:10.4213/mzm9891
[47] A. M. Selitskii, ”Modeling some optical systems on the basis of a parabolic differential-difference equation,” Matem. Modelirovanie, 24:12 (2012), 38–42. · Zbl 1313.78003
[48] R. V. Shamin, ”Spaces of initial data for differential equations in a Hilbert space,” Mat. Sb., 194:9 (2003), 141–156; English transl.: Russian Acad. Sci. Sb. Math., 194:9 (2003), 1411–1426. · Zbl 1073.34070 · doi:10.4213/sm770
[49] I. Ya. Shneiberg, ”Spectral properties of linear operators in interpolation families of Banach spaces,” Mat. Issled., 9:2 (1974), 214–229.
[50] A. L. Skubachevskii and R. V. Shamin, ”Second-order parabolic differential-difference equations,” Dokl. Ross. Akad. Nauk, 379:5 (2001), 595–598. · Zbl 1048.35122
[51] A. L. Skubachevskii and R. V. Shamin, ”The mixed boundary value problem for parabolic differential-difference equation,” Funct. Differ. Eq., 8:3–4 (2001), 407–424. · Zbl 1054.35119
[52] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Publ. Co., Amsterdam-New York, 1978. · Zbl 0387.46032
[53] M. I. Vishik, ”On strongly elliptic systems of differential equations,” Mat. Sb., 29(71):3 (1951), 615–676.
[54] T. W. Wolff, ”A note on interpolation spaces,” in: Lecture Notes in Math., vol. 918, Springer-Verlag, Berlin-New York, 1982, 199–204.
[55] A. Yagi, ”Coincidence entre des espaces d’interpolation et des domaines de puissances fractionaires d’opŕeateurs,” C. R. Acad. Sci. Paris, Sér. 1, 299:6 (1984), 173–176. · Zbl 0563.46042
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