Abstract
The paper deals with Green’s matrix of the Dirichlet problem forthe Stokes system in a polyhedron. In particular, Höolder estimates for thederivatives of the elements of this matrix are obtained in the case where thepolyhedronis convex.
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References
Dauge, M.: Stationary Stokes and Navier-Stokes systems on two- and three-dimensional with corners. Part 1: Linearized equations. SIAM J. Math. Anal. 20, 74–97 (1989)
Guzmán, J., Leykekhman, D., Rossmann, J., Schatz, A.-H.: Hölder estimates for Green's functions on convex polyhedral domains and their applications to finite element methods. Numer. Math. 112, 221–243 (2009)
Kozlov, V.A., Maz'ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Am. Math. Soc., Providence, RI (2001)
Kozlov, V.A., Maz'ya, V.G., Schwab, C.: On singularities of solutions to the Dirichlet problem of hydrodynamics near the vertex of a cone. J. Rein. Angew. Math. 456, 65–97 (1994)
Maz'ya, V.G., Plamenevskii, B.A.: Estimates of Green's functions and Schauder estimates for solutions of elliptic boundary value problems in a dihedral angle (Russian). Sib. Mat. Zh. 19, no. 5, 1064–1082 (1978); English transl.: Sib. Math. J. 19, 752–764 (1979)
Maz'ya, V.G., Plamenevskii, B.A.: Schauder estimations of solutions of elliptic boundary value problems in domains with edges on the boundary (Russian). Tr. Semin. S.L. Soboleva 1978, No. 2, 69–102 (1978); English transl.: Am. Math. Soc. Transl. 123, 141–169 (1984)
Maz'ya, V.G., Plamenevskii, B.A.: On the asymptotics of the fundamental solutions of elliptic boundary value problems in regtions with conical points (Russian). Probl. Mat. Anal. 7, 100–145 (1979); English transl.: Sel. Math. Sov. 4 363–397 (1985)
Maz'ya, V.G., Plamenevskii, B.A.: On properties of solutions of three-dimensional problems of elasticity theory and hydrodynamics in domains with isolated singular points (Russian). Dinam. Sploshn. Sred. 50, 99–120 (1981); English transl.: Am. Math. Soc. Transl. (2) 123, 109–123 (1984).
Maz'ya, V.G., Plamenevskii, B.A.: The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries. Part 1: Z. Anal. Anwend. 2, no. 4, 335–359 (1983); Part 2. ibid 2, no. 6, 523–551 (1983)
Maz'ya, V.G., Roßmann (Rossmann), J.: Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138, 27–53 (1988)
Maz'ya, V.G., Roßmann (Rossmann), J.: On the Agmon–Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Global Anal. Geom. 9, 253–303 (1991)
Maz'ya, V.G., Rossmann, J.: Point estimates for Green's matrix to boundary value problems for second order elliptic systems in a polyhedral cone. Z. Angew. Math. Mech. 82, no. 5, 291–316 (2002)
Maz'ya, V.G., Rossmann, J.: Pointwise estimates for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone. Math. Nachr. 278, no. 15, 1766–1810 (2005)
Maz'ya, V.G., Rossmann, J.: L p estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Math. Nachr. 280, no. 7, 751–793 (2007)
Maz'ya, V.G., Rossmann, J.: Mixed boundary value problems for the Navier-Stokes system in polyhedral domains. Arch. Ration. Mech. Anal. [To appear]
Rossmann, J.: On two classes of weighted Sobolev-Slobodetskiĭ spaces in a dihedral angle. Partial Differential Equations, Branch Center Publications 27, 399–424 (1992)
Verzhbinskii, G.M., Maz'ya, V.G.: On the asymptotics of solutions of the Dirichlet problem near a nonregular boundary. Dokl. Akad. Nauk SSSR 176, no. 3, 498–501 (1967)
Verzhbinskii, G.M., Maz'ya, V.G.: Asymptotic behaviour of solutions of elliptic equations of second order near the boundary I, II (Russian). Sib. Mat. Zh. 12, no. 6, 1217–1249, (1971); ibid 13, no. 6, 1239–1271 (1972)
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Roßmann, J. (2010). Hölder Estimates for Green’s Matrix of the Stokes System in Convex Polyhedra. In: Laptev, A. (eds) Around the Research of Vladimir Maz'ya II. International Mathematical Series, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1343-2_15
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DOI: https://doi.org/10.1007/978-1-4419-1343-2_15
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