Boundary value problems for the Stokes system in arbitrary Lipschitz domains

Mitrea, Marius ; Wright, Matthew

Astérisque, no. 344 (2012) , 247 p.

MR Zbl

@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},
     mrnumber = {2987056},
     zbl = {1345.35076},
     language = {en},
     url = {https://numdam.org/item/AST_2012__344__R3_0/}
}

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AU  - Wright, Matthew
TI  - Boundary value problems for the Stokes system in arbitrary Lipschitz domains
T3  - Astérisque
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%0 Book
%A Mitrea, Marius
%A Wright, Matthew
%T Boundary value problems for the Stokes system in arbitrary Lipschitz domains
%S Astérisque
%D 2012
%N 344
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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/

Bibliographie
Cité par

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

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

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | EuDML

[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 | Zbl

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | DOI

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

[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. | MR | Zbl | EuDML | Numdam

[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. | MR | Zbl | DOI

[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. | MR | Zbl | DOI

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

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

[17] B. E. J. Dahlberg, " $L^{q}$ -estimates for Green potentials in Lipschitz domains", Math. Scand. 44 (1979), p. 149-170. | MR | Zbl | EuDML | DOI

[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. | MR | Zbl | DOI

[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 | Zbl

[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. | MR | Zbl | EuDML | Numdam | DOI

[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 | Zbl

[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. | MR | Zbl | DOI

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

[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. | MR | Zbl | DOI

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

[26] P. Deuring - "The Stokes system in $3 D$ -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 | Zbl

[27] P. Deuring, "The $3 D$ 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. | MR | Zbl | DOI

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

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

[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. | MR | Zbl | DOI

[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 | Zbl

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

[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 | Zbl

[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. | MR | Zbl | DOI

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

[37] Y. Giga - "Analyticity of the semigroup generated by the Stokes operator in $L_{r}$ spaces", Math. Z. 178 (1981), p. 297-329. | MR | Zbl | EuDML | DOI

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

[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. | MR | Zbl | DOI

[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 | Zbl

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

[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. | MR | Zbl | DOI

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

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | DOI

[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 | Zbl

[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. | MR | Zbl | DOI

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

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | EuDML

[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 | Zbl

[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. | MR | Zbl | EuDML

[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. | MR | Zbl | DOI

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

[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 | Zbl

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

[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 | Zbl

[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. | MR | Zbl | EuDML | DOI

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

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

[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 | Zbl

[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. | MR | Zbl | DOI

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

[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. | MR | Zbl | DOI

[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. | MR | Zbl | EuDML | DOI

[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. | MR | Zbl | DOI

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

[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. | MR | JFM | EuDML | DOI

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

[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 | Zbl

[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. | MR | Zbl | DOI

[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. | MR | Zbl | DOI

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

[79] Z. Shen, "The $L^{p}$ boundary value problems on Lipschitz domains", Adv. Math. 216 (2007), p. 212-254. | MR | Zbl | DOI

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

[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 | Zbl

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

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

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

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

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

[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. | MR | Zbl | EuDML | DOI

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

[89] N. T. Varopoulos - "Zeros of $H^{p}$ functions in several complex variables", Pacific J. Math. 88 (1980), p. 189-246. | MR | Zbl | DOI

[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. | MR | Zbl | DOI

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