Fractional Powers of Operators Corresponding to Coercive Problems in Lipschitz Domains

M. S. Agranovich ; A. M. Selitskii

Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 2, pp. 2-17.

Résumé

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\ge2$, and let $L$ be a second-order matrix strongly elliptic operator in $\Omega$ written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation $Lu=f$, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well. We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary $\Gamma=\partial\Omega$ or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem.

Détail

Keywords: Lipschitz domain, strongly elliptic system, coercive problem, Kato's square root problem.

Bibliographie

[1] M. C. Agranovich, “Regulyarnost variatsionnykh reshenii lineinykh granichnykh zadach v lipshitsevykh oblastyakh”, Funkts. analiz i ego pril., 40:4 (2006), 83–103 | DOI | MR | Zbl

[2] M. C. Agranovich, “K teorii zadach Dirikhle i Neimana dlya lineinykh silno ellipticheskikh sistem v lipshitsevykh oblastyakh”, Funkts. analiz i ego pril., 41:4 (2007), 1–21 | DOI | MR | Zbl

[3] M. C. Agranovich, “Operatory tipa potentsiala i zadachi sopryazheniya dlya silno ellipticheskikh sistem 2-go poryadka v oblastyakh s lipshitsevoi granitsei”, Funkts. analiz i ego pril., 43:3 (2009), 3–25 | DOI | MR | Zbl

[4] M. C. Agranovich, “Silno ellipticheskie sistemy 2-go poryadka s granichnymi usloviyami na nezamknutoi lipshitsevoi poverkhnosti”, Funkts. analiz i ego pril., 45:1 (2011), 1–15 | DOI | MR | Zbl

[5] M. C. Agranovich, “Smeshannye zadachi v lipshitsevoi oblasti dlya silno ellipticheskikh sistem 2-go poryadka”, Funkts. analiz i ego pril., 45:2 (2011), 1–22 | DOI | MR | Zbl

[6] M. S. Agranovich, “Remarks on strongly elliptic second-order systems in Lipschitz domains”, Russian J. Math. Phys., 20:4 (2012), 405–416 | DOI | MR

[7] M. C. Agranovich, Sobolevskie prostranstva, ikh obobscheniya i ellipticheskie zadachi v oblastyakh s gladkoi i lipshitsevoi granitsei, Izd. MTsNMO, M., 2013

[8] M. C. Agranovich, M. I. Vishik, “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, 19:3 (1964), 53–161 | MR | Zbl

[9] I. Berg, I. Lëfstrëm, Interpolyatsionnye prostranstva, Mir, M., 1980 | MR

[10] M. I. Vishik, “O silno ellipticheskikh sistemakh differentsialnykh uravnenii”, Matem. sb., 29(71):3 (1951), 615–676 | Zbl

[11] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[12] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, P. E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR

[13] Zh.-L. Lions, E. Madzhenes, Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[14] A. M. Selitskii, “Prostranstvo nachalnykh dannykh 3-i kraevoi zadachi dlya parabolicheskogo differentsialno-raznostnogo uravneniya v odnomernom sluchae”, Matem. zametki, 92:4 (2012), 636–640 | DOI | MR | Zbl

[15] A. M. Selitskii, “Modelirovanie nekotorykh opticheskikh sistem na osnove parabolicheskogo differentsialno-raznostnogo uravneniya”, Matem. modelirovanie, 24:12 (2012), 38–42 | MR | Zbl

[16] A. L. Skubachevskii, R. V. Shamin, “Parabolicheskie differentsialno-raznostnye uravneniya vtorogo poryadka”, Dokl. RAN, 379:5 (2001), 595–598 | MR | Zbl

[17] Kh. Tribel, Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | MR | Zbl

[18] L. Khërmander, Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, M., 1965 | MR

[19] R. V. Shamin, “O prostranstvakh nachalnykh dannykh dlya differentsialnykh uravnenii v gilbertovom prostranstve”, Matem. sb., 194:9 (2003), 141–156 | DOI | MR | Zbl

[20] I. Ya. Shneiberg, “Spektralnye svoistva lineinykh operatorov v interpolyatsionnykh semeistvakh banakhovykh prostranstv”, Matem. issled., 9:2 (1974), 214–229 | MR | Zbl

[21] Sh. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems”, Comm. Pure Appl. Math., 15:2 (1962), 119–147 | DOI | MR | Zbl

[22] W. Arendt, “Semigroups and evolution equations: functional calculus, regularity and kernel estimates”, Handbook of Differential Equations, Evolutionary Differential Equations, v. 1, Elsevier/North-Holland, Amsterdam, 2004, 1–85 | MR | Zbl

[23] P. Auscher, N. Badr, R. Haller-Dintelmann, J. Rehberg, The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$, arXiv: 1210.0780v1

[24] P. Auscher, S. Hofmann, M. Lacey, J. Lewis, A. McIntosh, P. Tchamitchian, “The solution of Kato's conjectures”, C. R. Acad. Sci. Paris, Sér. 1, 332:7 (2001), 601–606 | DOI | MR | Zbl

[25] P. Auscher, A. McIntosh, A. Nahmod, “Holomorphic functional calculi of operators, quadratic estimates and interpolation”, Indiana Univ. Math. J., 46:2 (1997), 375–403 | DOI | MR | Zbl

[26] P. Auscher, S. Hofmann, A. McIntosch, 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 | DOI | MR | Zbl

[27] P. Auscher, P. Tchamitchian, “Square root problem for divergence operators and related topics”, Astérisque, 249 (1998), 1–171 | MR

[28] P. Auscher, P. Tchamitchian, “Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^2$ theory”, J. Anal. Math., 90 (2003), 1–12 | DOI | MR | Zbl

[29] P. Auscher, P. Tchamitchian, “Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^p$ theory”, Math. Ann., 320:3 (2001), 577–623 | DOI | MR | Zbl

[30] A. Axelsson, S. Keith, A. McIntosh, “The Kato square root problem for mixed boundary value problems”, J. London Math. Soc., 74:1 (2006), 113–130 | DOI | MR | Zbl

[31] S. Blunck, P. Kunstmann, “Calderón–Zygmund theory for non-integral operators and the $H^\infty$ functional calculus”, Rev. Mat. Iberoamericana, 19:3 (2003), 919–942 | DOI | MR | Zbl

[32] A. F. M. ter Elst, D. W. Robinson, “On Kato's square root problem”, Hokkaido Math. J., 26:2 (1997), 365–376 | MR | Zbl

[33] J. Griepentrog, K. Gröger, H.-Ch. Kaiser, J. Rehberg, “Interpolation for function spaces related to mixed boundary value problems”, Math. Nachr., 241 (2002), 110–120 | 3.0.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[34] D. Grisvard, “Caractérisation de quelques espaces d'interpolation”, Arc. Rational Mech. Anal., 25 (1967), 40–63 | DOI | MR | Zbl

[35] M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006 | MR | Zbl

[36] S. Hoffmann, “A short course on the Kato problem”, Contemp. Math., 289 (2001), 61–67 | DOI | MR

[37] T. Hytönen, A. McIntosh, P. Portal, “Kato's square root problem in Banach spaces”, J. Funct. Anal., 254:3 (2008), 675–726 | DOI | MR | Zbl

[38] S. Janson, P. Nilsson, J. Peetre, “Notes on Wolff's note on interpolation spaces”, Proc. London Math. Soc. (3), 48:2 (1984), 283–299 | DOI | MR | Zbl

[39] T. Kato, “Fractional powers of dissipative operators”, J. Math. Soc. Japan, 13 (1961), 246–274 | DOI | MR | Zbl

[40] T. Kato, “Fractional powers of dissipative operators, II”, J. Math. Soc. Japan, 14 (1962), 242–248 | DOI | MR | Zbl

[41] H. Komatsu, “Fractional powers of operators”, Pacif. J. Math., 19 (1966), 285–346 | DOI | MR | Zbl

[42] J. L. Lions, Équations différentielles operationelles et problèmes aux limites, Springer-Verlag, Berlin etc., 1961 | MR | Zbl

[43] J. L. Lions, “Espaces d'interpolation et domaines de puissances fractionaires d'opeŕateurs”, J. Math. Soc. Japan, 14 (1962), 233–241 | DOI | MR | Zbl

[44] A. McIntosh, “On the compatibility of $A^{1/2}$ and $A^{*1/2}$”, Proc. Amer. Math. Soc., 32:2 (1972), 430–434 | MR | Zbl

[45] A. McIntosh, “Square roots of elliptic operators”, J. Funct. Anal., 61:3 (1985), 307–327 | DOI | MR | Zbl

[46] A. McIntosh, “Operators which have an $H^{\infty}$ functional calculus”, Miniconference on Operator Theory and Partial Differential Equations, Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, 1986, 210–231 | MR | Zbl

[47] A. McIntosh, The Square Root Problem for Elliptic Operators: a Survey, Lecture Notes in Math., 1450, Springer-Verlag, Berlin, 1990, 122–140 | DOI | MR

[48] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge, UK, 2000 | MR

[49] J. Nec̆as, Les méthodes directes en théorie des eq́uations elliptiques, Masson, Paris, 1967 ; Direct Methods in the Theory of Elliptic Equations, Springer-Verlag, Berlin–Heidelberg, 2012 | MR | Zbl | MR

[50] L. Nirenberg, “Remarks on strongly elliptic partial differential equations”, Comm. Pure Appl. Math., 8 (1965), 649–675 | MR

[51] R. T. Seeley, “Norms and domains of the complex powers $A_B^z$”, Amer. J. Math., 93:2 (1971), 299–309 | DOI | MR | Zbl

[52] R. T. Seeley, “Interpolation in $L_p$ with boundary conditions”, Studia Math., 44 (1972), 47–60 | DOI | MR | Zbl

[53] A. L. Skubachevskii, R. V. Shamin, “The mixed boundary value problem for parabolic differential-difference equation”, Funct. Differ. Eq., 8:3–4 (2001), 407–424 | MR | Zbl

[54] T. W. Wolff, “A note on interpolation spaces”, Lecture Notes in Math., 918, Springer-Verlag, Berlin–New York, 1982, 199–204 | DOI | MR

[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 | MR | Zbl