June  2011, 15(4): 999-1018. doi: 10.3934/dcdsb.2011.15.999

Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds

1. 

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca

2. 

Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart

Received  January 2010 Published  March 2011

In this paper we use a layer potential method to obtain an existence and solvability result in Sobolev spaces for a Dirichlet-transmission problem given in terms of general Brinkman operators, when the solution is defined in two adjacent Lipschitz or $C^1$ domains on a Riemannian manifold and satisfies prescribed transmission conditions at the interface between these domains, and an additional Dirichlet condition on an external boundary.
Citation: Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 999-1018. doi: 10.3934/dcdsb.2011.15.999
References:
[1]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results,, SIAM J. Math. Anal., 19 (1988), 613. doi: doi:10.1137/0519043. Google Scholar

[2]

M. Dindoš, Hardy spaces and potential theory on $C^1$ domains in Riemannian manifolds,, Memoirs of the Amer. Math. Soc., 191 (2008). Google Scholar

[3]

M. Dindoš and 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), 1. doi: doi:10.1007/s00205-004-0320-y. Google Scholar

[4]

L. Escauriaza and M. Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains,, J. Funct. Anal., 216 (2004), 141. doi: doi:10.1016/j.jfa.2003.12.005. Google Scholar

[5]

E. Fabes, M. Jodeit and N. Rivère, Potential techniques for boundary value problems on $C^1$-domains,, Acta Math., 141 (1978), 165. doi: doi:10.1007/BF02545747. Google Scholar

[6]

E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains,, Duke Math. J., 57 (1988), 769. doi: doi:10.1215/S0012-7094-88-05734-1. Google Scholar

[7]

E. Fabes, O. Mendez and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains,, J. Funct. Anal., 159 (1998), 323. doi: doi:10.1006/jfan.1998.3316. Google Scholar

[8]

A. N. Filippov, S. I. Vasin and V. M. Starov, Mathematical modeling of the hydrodynamic permeability of a membrane built up from porous particles with a permeable shell,, Colloids and Surfaces A, 282-283 (2006), 282. doi: i:10.1016/j.colsurfa.2005.12.001. Google Scholar

[9]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 79 (2008), 105. Google Scholar

[10]

G. C. Hsiao and W. L. Wendland, "Boundary Integral Equations,", Springer, (2008). doi: doi:10.1007/978-3-540-68545-6. Google Scholar

[11]

M. Kohr, C. Pintea and W. L. Wendland, Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds,, Commun. Pure Appl. Anal., 9 (2010), 493. doi: doi:10.3934/cpaa.2010.9.493. Google Scholar

[12]

M. Kohr, C. Pintea and W. L. Wendland, Brinkman-type operators on Riemannian manifolds: Transmission problems in Lipschitz and $C^1$ domains,, Potential Anal., 32 (2010), 229. doi: doi:10.1007/s11118-009-9151-7. Google Scholar

[13]

M. Kohr, C. Pintea and W. L. Wendland, On mapping properties of layer potential operators for Brinkman equations on Lipschitz domains in Riemannian manifolds,, Mathematica (Cluj), 52 (2010), 31. Google Scholar

[14]

M. Kohr and I. Pop, "Viscous Incompressible Flow for Low Reynolds Numbers,", WIT Press, (2004). Google Scholar

[15]

M. Kohr, G. P. Raja Sekhar and J. R. Blake, Green's function of the Brinkman equation in a 2D anisotropic case,, IMA J. Appl. Math., 73 (2008), 374. doi: doi:10.1093/imamat/hxm023. Google Scholar

[16]

M. Kohr, G. P. Raja Sekhar and W. L. Wendland, Boundary integral equations for a three-dimensional Stokes-Brinkman cell model,, Math. Models Methods Appl. Sci., 18 (2008), 2055. doi: doi:10.1142/S0218202508003297. Google Scholar

[17]

M. Kohr and W. L. Wendland, Boundary integral equations for a three-dimensional Brinkman flow problem,, Math. Nachr., 282 (2009), 1305. doi: doi:10.1002/mana.200710797. Google Scholar

[18]

H. B. Jr. Lawson and M-L. Michelsohn, "Spin Geometry,", Princeton University Press, (1989). Google Scholar

[19]

V. Maz'ya, M. Mitrea and T. Shaposhnikova, The inhomogeneous Dirichlet problem for the Stokes system in Lipschitz domains with unit normal close to VMO,, Funct. Anal. Appl., 43 (2009), 217. doi: doi:10.1007/s10688-009-0029-7. Google Scholar

[20]

S. E. Mikhailov, Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients,, preprint, (). Google Scholar

[21]

D. Mitrea, M. Mitrea and Shi Qiang, Variable coefficient transmission problems and singular integral operators on non-smooth manifolds,, J. Integral Equations Appl., 18 (2006), 361. doi: doi:10.1216/jiea/1181075395. Google Scholar

[22]

D. Mitrea, M. Mitrea and M. Taylor, Layer potentials, the Hodge Laplacian and global boundary problems in non-smooth Riemannian manifolds,, Memoirs of the Amer. Math. Soc., 150 (2001). Google Scholar

[23]

M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds,, J. Funct. Anal., 163 (1999), 181. doi: doi:10.1006/jfan.1998.3383. Google Scholar

[24]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Hölder continuous metric tensors,, Comm. Partial Differential Equations, 25 (2000), 1487. doi: doi:10.1080/03605300008821557. Google Scholar

[25]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem,, J. Funct. Anal., 176 (2000), 1. doi: doi:10.1006/jfan.2000.3619. Google Scholar

[26]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Lp, Hardy and Hölder type results,, Comm. Anal. Geom., 9 (2001), 369. Google Scholar

[27]

M. Mitrea and M. Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds,, Math. Anal., 321 (2001), 955. doi: doi:10.1007/s002080100261. Google Scholar

[28]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: the case of Dini metric tensors,, Trans. Amer. Math. Soc., 355 (2003), 1961. doi: doi:10.1090/S0002-9947-02-03150-1. Google Scholar

[29]

M. Mitrea and M. Taylor, Sobolev and Besov space estimates for solutions to second order PDE on Lipschitz domains in manifolds with Dini or Hölder continuous metric tensors,, Comm. Partial Differential Equations, 30 (2005), 1. doi: doi:10.1081/PDE-200044425. Google Scholar

[30]

M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains,, Ast$\acutee$risque, (). Google Scholar

[31]

M. Taylor, "Pseudodifferential Operators,", Princeton University Press, (1981). Google Scholar

[32]

M. Taylor, "Partial Differential Equations,", Springer-Verlag, 1-3 (): 1996. Google Scholar

[33]

J. T. Wloka, B. Rowley and B. Lawruk, "Boundary Value Problems for Elliptic Systems,", Cambridge University Press, (1995). doi: doi:10.1017/CBO9780511662850. Google Scholar

show all references

References:
[1]

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results,, SIAM J. Math. Anal., 19 (1988), 613. doi: doi:10.1137/0519043. Google Scholar

[2]

M. Dindoš, Hardy spaces and potential theory on $C^1$ domains in Riemannian manifolds,, Memoirs of the Amer. Math. Soc., 191 (2008). Google Scholar

[3]

M. Dindoš and 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), 1. doi: doi:10.1007/s00205-004-0320-y. Google Scholar

[4]

L. Escauriaza and M. Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains,, J. Funct. Anal., 216 (2004), 141. doi: doi:10.1016/j.jfa.2003.12.005. Google Scholar

[5]

E. Fabes, M. Jodeit and N. Rivère, Potential techniques for boundary value problems on $C^1$-domains,, Acta Math., 141 (1978), 165. doi: doi:10.1007/BF02545747. Google Scholar

[6]

E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains,, Duke Math. J., 57 (1988), 769. doi: doi:10.1215/S0012-7094-88-05734-1. Google Scholar

[7]

E. Fabes, O. Mendez and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains,, J. Funct. Anal., 159 (1998), 323. doi: doi:10.1006/jfan.1998.3316. Google Scholar

[8]

A. N. Filippov, S. I. Vasin and V. M. Starov, Mathematical modeling of the hydrodynamic permeability of a membrane built up from porous particles with a permeable shell,, Colloids and Surfaces A, 282-283 (2006), 282. doi: i:10.1016/j.colsurfa.2005.12.001. Google Scholar

[9]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains,, in, 79 (2008), 105. Google Scholar

[10]

G. C. Hsiao and W. L. Wendland, "Boundary Integral Equations,", Springer, (2008). doi: doi:10.1007/978-3-540-68545-6. Google Scholar

[11]

M. Kohr, C. Pintea and W. L. Wendland, Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds,, Commun. Pure Appl. Anal., 9 (2010), 493. doi: doi:10.3934/cpaa.2010.9.493. Google Scholar

[12]

M. Kohr, C. Pintea and W. L. Wendland, Brinkman-type operators on Riemannian manifolds: Transmission problems in Lipschitz and $C^1$ domains,, Potential Anal., 32 (2010), 229. doi: doi:10.1007/s11118-009-9151-7. Google Scholar

[13]

M. Kohr, C. Pintea and W. L. Wendland, On mapping properties of layer potential operators for Brinkman equations on Lipschitz domains in Riemannian manifolds,, Mathematica (Cluj), 52 (2010), 31. Google Scholar

[14]

M. Kohr and I. Pop, "Viscous Incompressible Flow for Low Reynolds Numbers,", WIT Press, (2004). Google Scholar

[15]

M. Kohr, G. P. Raja Sekhar and J. R. Blake, Green's function of the Brinkman equation in a 2D anisotropic case,, IMA J. Appl. Math., 73 (2008), 374. doi: doi:10.1093/imamat/hxm023. Google Scholar

[16]

M. Kohr, G. P. Raja Sekhar and W. L. Wendland, Boundary integral equations for a three-dimensional Stokes-Brinkman cell model,, Math. Models Methods Appl. Sci., 18 (2008), 2055. doi: doi:10.1142/S0218202508003297. Google Scholar

[17]

M. Kohr and W. L. Wendland, Boundary integral equations for a three-dimensional Brinkman flow problem,, Math. Nachr., 282 (2009), 1305. doi: doi:10.1002/mana.200710797. Google Scholar

[18]

H. B. Jr. Lawson and M-L. Michelsohn, "Spin Geometry,", Princeton University Press, (1989). Google Scholar

[19]

V. Maz'ya, M. Mitrea and T. Shaposhnikova, The inhomogeneous Dirichlet problem for the Stokes system in Lipschitz domains with unit normal close to VMO,, Funct. Anal. Appl., 43 (2009), 217. doi: doi:10.1007/s10688-009-0029-7. Google Scholar

[20]

S. E. Mikhailov, Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients,, preprint, (). Google Scholar

[21]

D. Mitrea, M. Mitrea and Shi Qiang, Variable coefficient transmission problems and singular integral operators on non-smooth manifolds,, J. Integral Equations Appl., 18 (2006), 361. doi: doi:10.1216/jiea/1181075395. Google Scholar

[22]

D. Mitrea, M. Mitrea and M. Taylor, Layer potentials, the Hodge Laplacian and global boundary problems in non-smooth Riemannian manifolds,, Memoirs of the Amer. Math. Soc., 150 (2001). Google Scholar

[23]

M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds,, J. Funct. Anal., 163 (1999), 181. doi: doi:10.1006/jfan.1998.3383. Google Scholar

[24]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Hölder continuous metric tensors,, Comm. Partial Differential Equations, 25 (2000), 1487. doi: doi:10.1080/03605300008821557. Google Scholar

[25]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem,, J. Funct. Anal., 176 (2000), 1. doi: doi:10.1006/jfan.2000.3619. Google Scholar

[26]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Lp, Hardy and Hölder type results,, Comm. Anal. Geom., 9 (2001), 369. Google Scholar

[27]

M. Mitrea and M. Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds,, Math. Anal., 321 (2001), 955. doi: doi:10.1007/s002080100261. Google Scholar

[28]

M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: the case of Dini metric tensors,, Trans. Amer. Math. Soc., 355 (2003), 1961. doi: doi:10.1090/S0002-9947-02-03150-1. Google Scholar

[29]

M. Mitrea and M. Taylor, Sobolev and Besov space estimates for solutions to second order PDE on Lipschitz domains in manifolds with Dini or Hölder continuous metric tensors,, Comm. Partial Differential Equations, 30 (2005), 1. doi: doi:10.1081/PDE-200044425. Google Scholar

[30]

M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains,, Ast$\acutee$risque, (). Google Scholar

[31]

M. Taylor, "Pseudodifferential Operators,", Princeton University Press, (1981). Google Scholar

[32]

M. Taylor, "Partial Differential Equations,", Springer-Verlag, 1-3 (): 1996. Google Scholar

[33]

J. T. Wloka, B. Rowley and B. Lawruk, "Boundary Value Problems for Elliptic Systems,", Cambridge University Press, (1995). doi: doi:10.1017/CBO9780511662850. Google Scholar

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