Boundary value problems for the Stokes system in arbitrary Lipschitz domains
Astérisque, no. 344 (2012) , 247 p.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez le site de la revue.
@book{AST_2012__344__R3_0,
     author = {Mitrea, Marius and Wright, Matthew},
     title = {Boundary value problems for the {Stokes} system in arbitrary {Lipschitz} domains},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {344},
     year = {2012},
     zbl = {1345.35076},
     mrnumber = {2987056},
     language = {en},
     url = {http://www.numdam.org/item/AST_2012__344__R3_0/}
}
TY  - BOOK
AU  - Mitrea, Marius
AU  - Wright, Matthew
TI  - Boundary value problems for the Stokes system in arbitrary Lipschitz domains
T3  - Astérisque
PY  - 2012
DA  - 2012///
IS  - 344
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2012__344__R3_0/
UR  - https://zbmath.org/?q=an%3A1345.35076
UR  - https://www.ams.org/mathscinet-getitem?mr=2987056
LA  - en
ID  - AST_2012__344__R3_0
ER  - 
Mitrea, Marius; Wright, Matthew. Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque, no. 344 (2012), 247 p. http://numdam.org/item/AST_2012__344__R3_0/

[1] V. Adolfsson & J. Pipher - "The inhomogeneous Dirichlet problem for Δ 2 in Lipschitz domains", J. Funct. Anal. 159 (1998), p. 137-190. | Article | MR 1654182 | Zbl 0934.35038

[2] H. Aimar - "Singular integrals and approximate identities on spaces of homogeneous type", Trans. Amer. Math. Soc. 292 (1985), p. 135-153. | Article | MR 805957 | Zbl 0578.42016

[3] E. Albrecht & V. Müller - "Spectrum of interpolated operators", Proc. Amer. Math. Soc. 129 (2001), p. 807-814. | Article | MR 1804050 | Zbl 0967.46021

[4] C. Amrouche & V. Girault - "On the existence and regularity of the solution of Stokes problem in arbitrary dimension", Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), p. 171-175. | Article | MR 1114965 | Zbl 0752.35047

[5] C. Amrouche & V. Girault, "Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension", Czechoslovak Math. J. 44 (119) (1994), p. 109-140. | EuDML 31399 | MR 1257940 | Zbl 0823.35140

[6] M. E. Bogovskiĭ - "Solutions of some problems of vector analysis, associated with the operators div and grad", in Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. , vol. 1980, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., 1980, p. 5-40, 149. | MR 631691 | Zbl 0479.58018

[7] M. Bownik - "Boundedness of operators on Hardy spaces via atomic decompositions", Proc. Amer. Math. Soc. 133 (2005), p. 3535-3542. | Article | MR 2163588 | Zbl 1070.42006

[8] R. M. Brown - personal communication.

[9] R. M. Brown & Z. Shen - "Estimates for the Stokes operator in Lipschitz domains", Indiana Univ. Math. J. 44 (1995), p. 1183-1206. | Article | MR 1386766 | Zbl 0858.35098

[10] A. P. Calderón - "Boundary value problems for the Laplace equation in Lipschitzian domains", in Recent progress in Fourier analysis (El Escorial, 1983), North-Holland Math. Stud., vol. 111, North-Holland, 1985, p. 33-48. | Article | MR 848140 | Zbl 0608.31001

[11] W. Cao & Y. Sagher - "Stability of Fredholm properties on interpolation scales", Ark. Mat. 28 (1990), p. 249-258. | Article | MR 1084014 | Zbl 0753.46036

[12] L. Cattabriga - "Su un problema al contorno relativo al sistema di equazioni di Stokes", Rend. Sem. Mat. Univ. Padova 31 (1961), p. 308-340. | EuDML 107065 | Numdam | MR 138894 | Zbl 0116.18002

[13] D.-C. Chang, S. G. Krantz & E. M. Stein - "Hardy spaces and elliptic boundary value problems", in The Madison Symposium on Complex Analysis (Madison, WI, 1991), Contemp. Math., vol. 137, Amer. Math. Soc., 1992, p. 119-131. | Article | MR 1190976 | Zbl 0765.42014

[14] D.-C. Chang, S. G. Krantz & E. M. Stein, "H p theory on a smooth domain in 𝐑 N and elliptic boundary value problems", J. Funct. Anal. 114 (1993), p. 286-347. | Article | MR 1223705 | Zbl 0804.35027

[15] R. R. Coifman & G. Weiss - "Extensions of Hardy spaces and their use in analysis", Bull. Amer. Math. Soc. 83 (1977), p. 569-645. | Article | MR 447954 | Zbl 0358.30023

[16] B. E. J. Dahlberg - "Estimates of harmonic measure", Arch. Rational Mech. Anal. 65 (1977), p. 275-288. | Article | MR 466593 | Zbl 0406.28009

[17] B. E. J. Dahlberg, " L q -estimates for Green potentials in Lipschitz domains", Math. Scand. 44 (1979), p. 149-170. | Article | EuDML 166638 | MR 544584 | Zbl 0418.31003

[18] B. E. J. Dahlberg & C. E. Kenig - "Hardy spaces and the Neumann problem in L p for Laplace's equation in Lipschitz domains", Ann. of Math. 125 (1987), p. 437-465. | Article | MR 890159 | Zbl 0658.35027

[19] B. E. J. Dahlberg & C. E. Kenig, "L p estimates for the three-dimensional systems of elastostatics on Lipschitz domains", in Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, 1990, p. 621-634. | MR 1044810 | Zbl 0702.35076

[20] B. E. J. Dahlberg, C. E. Kenig, J. Pipher & G. C. Verchota - "Area integral estimates for higher order elliptic equations and systems", Ann. Inst. Fourier (Grenoble) 47 (1997), p. 1425-1461. | Article | EuDML 75269 | Numdam | MR 1600375 | Zbl 0892.35053

[21] B. E. J. Dahlberg, C. E. Kenig & G. C. Verchota - "Boundary value problems for the systems of elastostatics in Lipschitz domains", Duke Math. J. 57 (1988), p. 795-818. | MR 975122 | Zbl 0699.35073

[22] M. Dauge - "Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations", SIAM J. Math. Anal. 20 (1989), p. 74-97. | Article | MR 977489 | Zbl 0681.35071

[23] R. Dautray & J.-L. Lions - Mathematical analysis and numerical methods for science and technology. Vol. 6, Springer, 1993. | MR 1295030 | Zbl 0802.35001

[24] W. J. Davis, D. J. H. Garling & N. Tomczak-Jaegermann - "The complex convexity of quasinormed linear spaces", J. Funct. Anal. 55 (1984), p. 110-150. | Article | MR 733036 | Zbl 0552.46012

[25] J. Detraz - "Propriétés de moyenne pour les solutions de systèmes elliptiques", Publ. Mat. 37 (1993), p. 83-89. | Article | EuDML 41510 | MR 1240924 | Zbl 0792.35046

[26] P. Deuring - "The Stokes system in 3D-Lipschitz domains: a survey of recent results", in Progress in partial differential equations: the Metz surveys, 4, Pitman Res. Notes Math. Ser., vol. 345, Longman, 1996, p. 187-204. | MR 1394714 | Zbl 0859.35089

[27] P. Deuring, "The 3D Stokes systems in domains with conical boundary points", in Theory of the Navier-Stokes equations, Ser. Adv. Math. Appl. Sci., vol. 47, World Sci. Publ., River Edge, NJ, 1998, p. 1-10. | Article | MR 1643021 | Zbl 0943.35064

[28] P. Deuring, "L p -theory for the Stokes system in 3D domains with conical boundary points", Indiana Univ. Math. J. 47 (1998), p. 11-47. | Article | MR 1631608 | Zbl 0914.76024

[29] M. Dindos & M. Mitrea - "Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains", Publ. Mat. 46 (2002), p. 353-403. | Article | EuDML 41457 | MR 1934200 | Zbl 1058.35073

[30] M. Dindos & M. Mitrea, "The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and C 1 domains", Arch. Ration. Mech. Anal. 174 (2004), p. 1-47. | Article | MR 2092995 | Zbl 1059.76014

[31] L. Escauriaza, M. Mitrea, J. Pipher & M. Wright - "The transmission problem for the Lame system in Lipschitz domains", preprint, 2009.

[32] E. B. Fabes, C. E. Kenig & G. C. Verchota - "The Dirichlet problem for the Stokes system on Lipschitz domains", Duke Math. J. 57 (1988), p. 769-793. | MR 975121 | Zbl 0685.35085

[33] M. Frazier & B. Jawerth - "A discrete transform and decompositions of distribution spaces", J. Funct. Anal. 93 (1990), p. 34-170. | Article | MR 1070037 | Zbl 0716.46031

[34] M. Frazier, B. Jawerth & G. Weiss - Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1991. | MR 1107300 | Zbl 0757.42006

[35] G. P. Galdi, C. G. Simader & H. Sohr - "On the Stokes problem in Lipschitz domains", Ann. Mat. Pura Appl. 167 (1994), p. 147-163. | Article | MR 1313554 | Zbl 0824.35098

[36] F. W. Gehring - "The L p -integrability of the partial derivatives of a quasicon-formal mapping", Acta Math. 130 (1973), p. 265-277. | Article | MR 402038 | Zbl 0258.30021

[37] Y. Giga - "Analyticity of the semigroup generated by the Stokes operator in L r spaces", Math. Z. 178 (1981), p. 297-329. | Article | EuDML 173103 | MR 635201 | Zbl 0473.35064

[38] V. Girault & P.-A. Raviart - Finite element approximation of the Navier- Stokes equations, Lecture Notes in Math., vol. 749, Springer, 1979. | MR 548867 | Zbl 0413.65081

[39] S. Hofmann, M. Mitrea & M. Taylor - "Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains", J. Geom. Anal. 17 (2007), p. 593-647. | Article | MR 2365661 | Zbl 1142.49021

[40] S. Hofmann, M. Mitrea & M. Taylor, "Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains", Int. Math. Res. Not. 2010 (2010), p. 2567-2865. | MR 2669659 | Zbl 1221.31010

[41] T. Iwaniec - "The Gehring lemma", in Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), Springer, 1998, p. 181-204. | Article | MR 1488451 | Zbl 0888.30017

[42] T. Jakab, I. Mitrea & M. Mitrea - "Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces", J. Funct Anal. 246 (2007), p. 50-112. | Article | MR 2316877 | Zbl 1162.42310

[43] D. Jerison & C. E. Kenig -"The inhomogeneous Dirichlet problem in Lipschitz domains", J. Funct Anal. 130 (1995), p. 161-219. | Article | MR 1331981 | Zbl 0832.35034

[44] N. J. Kalton - "Plurisubharmonic functions on quasi-Banach spaces", Studia Math. 84 (1986), p. 297-324. | Article | EuDML 218730 | MR 877084 | Zbl 0625.46021

[45] N. J. Kalton, S. Mayboroda & M. Mitrea - "Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations", in Interpolation theory and applications, Contemp. Math., vol. 445, Amer. Math. Soc., 2007, p. 121-177. | Article | MR 2381891 | Zbl 1158.46013

[46] N. J. Kalton & M. Mitrea - "Stability results on interpolation scales of quasi-Banach spaces and applications", Trans. Amer. Math. Soc. 350 (1998), p. 3903-3922. | Article | MR 1443193 | Zbl 0902.46002

[47] N. J. Kalton & M. Mitrea, in preparation.

[48] N. J. Kalton, N. T. Peck & J. W. Roberts - An F-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge Univ. Press, 1984. | MR 808777 | Zbl 0556.46002

[49] R. B. Kellogg & J. E. Osborn - "A regularity result for the Stokes problem in a convex polygon", J. Functional Analysis 21 (1976), p. 397-431. | Article | MR 404849 | Zbl 0317.35037

[50] J. Kilty - "The L p Dirichlet problem for the Stokes system on Lipschitz domains", Indiana Univ. Math. J. 58 (2009), p. 1219-1234. | Article | MR 2541365 | Zbl 1168.35033

[51] M. Kohr & I. Pop - Viscous incompressible flow for low Reynolds numbers, Advances in Boundary Elements, vol. 16, WIT Press, Southampton, 2004. | MR 2091145 | Zbl 1064.76001

[52] V. Kozlov & V. G. Maz'Ya - "Spectral properties of operator pencils generated by elliptic boundary value problems in a cone", Funktsional. Anal. i Prilozhen. 22 (1988), p. 38-46, 96, translation in Functional Anal Appl. 22 (1988), p. 114-121. | Article | MR 947604 | Zbl 0672.35050

[53] V. Kozlov & V. G. Maz'Ya, "On the spectrum of an operator pencil generated by the Dirichlet problem in a cone", Mat. Sb. 182 (1991), p. 638-660, translation in Math. USSR Sbornik 73 (1992), p. 27-48. | EuDML 72263 | MR 1124101 | Zbl 0782.35047

[54] V. Kozlov, V. G. Maz'Ya & J. Rossmann - Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, Amer. Math. Soc., 2001. | MR 1788991 | Zbl 0965.35003

[55] V. Kozlov, V. G. Maz'Ya & C. Schwab - "On singularities of solutions to the Dirichlet problem of hydrodynamics near the vertex of a cone", J. reine angew. Math. 456 (1994), p. 65-97. | EuDML 153662 | MR 1301452 | Zbl 0807.76017

[56] V. Kozlov & V. Maz'Ya - "Boundary behavior of solutions to linear and nonlinear elliptic equations in plane convex domains", Math. Res. Lett. 8 (2001), p. 189-193. | Article | MR 1825269 | Zbl 0987.35050

[57] R. A. Macías & C. Segovia - "Lipschitz functions on spaces of homogeneous type", Adv. in Math. 33 (1979), p. 257-270. | Article | MR 546295 | Zbl 0431.46018

[58] P. Maremonti, R. Russo & G. Starita - "On the Stokes equations: the boundary value problem", in Advances in fluid dynamics, Quad. Mat., vol. 4, Dept. Math., Seconda Univ. Napoli, Caserta, 1999, p. 69-140. | MR 1770189 | Zbl 0949.35111

[59] S. Mayboroda & M. Mitrea - "Sharp estimates for Green potentials on non-smooth domains", Math. Res. Lett. 11 (2004), p. 481-492. | Article | MR 2092902 | Zbl 1073.35075

[60] S. Mayboroda & M. Mitrea, "The solution of the Chang-Krein-Stein conjecture", in Proceedings of the Conference in Harmonic Analysis and its Applications, Tokyo Woman's Cristian University (A. Miyachi, ed.), 2007, p. 61-154.

[61] V. Maz'Ya & J. Rossmann - "Mixed boundary value problems for the Navier-Stokes system in polyhedral domains", preprint, 2008. | MR 2563641 | Zbl 1253.76019

[62] O. Mendez & M. Mitrea - "The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations", J. Fourier Anal. Appl. 6 (2000), p. 503-531. | Article | EuDML 59657 | MR 1781091 | Zbl 0972.46017

[63] Y. Meyer - Ondelettes et opérateurs. II, Actualités Mathématiques., Hermann, 1990. | Zbl 0745.42011

[64] I. Mitrea & M. Mitrea - "Multiple layer potentials for higher order elliptic boundary value problems", preprint, 2008. | MR 3013645

[65] I. Mitrea, M. Mitrea, J. Pipher & M. Wright - "Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains", preprint, 2008. | MR 2839870 | Zbl 1238.35024

[66] M. Mitrea & M. Taylor - "Potential theory on Lipschitz domains in Riemannian manifolds: L p Hardy, and Hölder space results", Comm. Anal. Geom. 9 (2001), p. 369-421. | Article | MR 1846208 | Zbl 0994.31001

[67] J. Necas - Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris, 1967. | MR 227584 | Zbl 1225.35003

[68] J. Pipher & G. C. Verchota - "The Dirichlet problem in L p for the biharmonic equation on Lipschitz domains", Amer. J. Math. 114 (1992), p. 923-972. | Article | MR 1183527 | Zbl 0768.31006

[69] J. Pipher & G. C. Verchota, "A maximum principle for biharmonic functions in Lipschitz and C 1 domains", Comment. Math. Helv. 68 (1993), p. 385-414. | Article | EuDML 140309 | MR 1236761 | Zbl 0794.31005

[70] J. Pipher & G. C. Verchota, "Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators", Ann. of Math. 142 (1995), p. 1-38. | Article | MR 1338674 | Zbl 0878.35035

[71] H. Power & L. C. Wrobel - Boundary integral methods in fluid mechanics, Computational Mechanics Publications, 1995. | Zbl 0815.76001

[72] H. Rademacher - "Über partielle und totale Differenzierbarkeit von Funktionen mehrerer Variablen und über die Transformation der Doppelintegrale", Math. Ann. 79 (1918), p. 340-359. | Article | EuDML 158802 | JFM 47.0243.01 | MR 1511935

[73] J. M. Rallison - "The deformation of small viscous drops and bubbles in shear flows", Ann. Rev. Fluid Mech. 16 (1984), p. 45-66. | Article | Zbl 0604.76079

[74] T. Runst & W. Sickel - Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., 1996. | MR 1419319 | Zbl 0873.35001

[75] V. S. Rychkov - "On restrictions and extensions of the Besov and Triebel- Lizorkin spaces with respect to Lipschitz domains", J. London Math. Soc. 60 (1999), p. 237-257. | Article | MR 1721827 | Zbl 0940.46017

[76] J. H. Shapiro - "The Riesz and Fredholm theories in linear topological spaces", preprint, 1995.

[77] Z. Shen - "A note on the Dirichlet problem for the Stokes system in Lipschitz domains", Proc. Amer. Math. Soc. 123 (1995), p. 801-811. | Article | MR 1223521 | Zbl 0816.35106

[78] Z. Shen, "The L p Dirichlet problem for elliptic systems on Lipschitz domains", Math. Res. Lett. 13 (2006), p. 143-159. | Article | MR 2200052 | Zbl 1174.35362

[79] Z. Shen, "The L p boundary value problems on Lipschitz domains", Adv. Math. 216 (2007), p. 212-254. | Article | MR 2353255 | Zbl 1210.35080

[80] Z. Slodkowski - "A generalization of Vesentini and Wermer's theorems", Rend. Sem. Mat. Univ. Padova 75 (1986), p. 157-171. | EuDML 108017 | Numdam | MR 847664 | Zbl 0629.47012

[81] I. J. Šneĭberg - "Spectral properties of linear operators in interpolation families of Banach spaces", Mat. Issled. 9 (1974), p. 214-229, 254-255. | MR 634681 | Zbl 0314.46033

[82] E. M. Stein - Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton Univ. Press, 1970. | MR 290095 | Zbl 0207.13501

[83] A. Tabacco Vignati & M. Vignati - "Spectral theory and complex interpolation", J. Funct. Anal. 80 (1988), p. 383-397. | Article | MR 961906 | Zbl 0676.46053

[84] R. Temam - Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., 1977, Studies in Mathematics and its Applications, Vol. 2. | MR 603444 | Zbl 0383.35057

[85] H. Triebel - Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, 1978. | MR 500580 | Zbl 0387.46033

[86] H. Triebel, Theory of function spaces, Monographs in Math., vol. 78, Birkhäuser, 1983. | MR 781540 | Zbl 0546.46027

[87] H. Triebel, "Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers", Rev. Mat. Complut. 15 (2002), p. 475-524. | Article | EuDML 44356 | MR 1951822 | Zbl 1034.46033

[88] W. Varnhorn - The Stokes equations, Mathematical Research, vol. 76, Akademie-Verlag, 1994. | MR 1282728 | Zbl 0813.35085

[89] N. T. Varopoulos - "Zeros of H p functions in several complex variables", Pacific J. Math. 88 (1980), p. 189-246. | Article | MR 595819 | Zbl 0454.32006

[90] G. C. Verchota - "Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains", J. Funct. Anal. 59 (1984), p. 572-611. | Article | MR 769382 | Zbl 0589.31005

[91] J. M. Wilson - "On the atomic decomposition for Hardy spaces", Pacific J. Math. 116 (1985), p. 201-207. | Article | MR 769832 | Zbl 0563.42012