# American Institute of Mathematical Sciences

January  2014, 13(1): 175-202. doi: 10.3934/cpaa.2014.13.175

## Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact 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  October 2012 Revised  April 2013 Published  July 2013

The aim of this paper is twofold. On the one hand we construct Neumann-transmission kernels for pseudodifferential Brinkman operators. They are used to provide simple representations of the solution to some transmission problems for the pseudodifferential Brinkman operator. On the other hand, we show the well-posedness of a Neumann-transmission problem for two pseudodifferential Brinkman operators on adjacent Lipschitz domains in a compact Riemannian manifold, with boundary data in some $L^p$, Sobolev or Besov spaces. We rely on the layer potential theory in order to obtain an explicit representation of the solution to this problem. Compactness and invertibility results of associated layer potential operators on $L^p$, Sobolev and Besov spaces are also presented.
Citation: Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact Riemannian manifolds. Communications on Pure & Applied Analysis, 2014, 13 (1) : 175-202. doi: 10.3934/cpaa.2014.13.175
##### References:
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Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math., 125 (1987), 437-465.  Google Scholar [7] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari (Italian), Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43.  Google Scholar [8] 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), 1-47. doi: 10.1007/s00205-004-0320-y.  Google Scholar [9] D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163.  Google Scholar [10] L. Escauriaza and M. Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains, J. Funct. Anal., 216 (2004), 141-171. doi: 10.1016/j.jfa.2003.12.005.  Google Scholar [11] E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793. doi: 10.1215/S0012-7094-88-05734-1.  Google Scholar [12] D. Fericean, Layer potential analysis of a Neumann problem for the Brinkman system,, Mathematica (Cluj), ().   Google Scholar [13] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 Edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar [14] S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172. doi: 10.1007/s00229-007-0107-1.  Google Scholar [15] S. Hofmann, M. Mitrea and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, Int. Math. Res. Notices, 14 (2010), 2567-2865. doi: 10.1093/imrn/rnp214.  Google Scholar [16] G. C. Hsiao and W. L. 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Wendland, Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains, Potential Anal., 38 (2013), 1123-1171. doi: 10.1007/s11118-012-9310-0.  Google Scholar [22] 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), 493-537. doi: 10.3934/cpaa.2010.9.493.  Google Scholar [23] 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), 229-273. doi: 10.1007/s11118-009-9151-7.  Google Scholar [24] M. Kohr, C. Pintea and W. L. Wendland, Dirichlet-transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 999-1018. doi: 10.3934/dcdsb.2011.15.999.  Google Scholar [25] M. Kohr, C. Pintea and W. L. Wendland, Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: Applications to pseudodifferential Brinkman operators, Int. Math. Res. Notices, 2012, DOI 10.1093/imrn/RNS158, to appear. doi: 10.1093/imrn/rns158.  Google Scholar [26] M. Kohr, C. Pintea and W. L. Wendland, Dirichlet-transmission problems for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in Riemannian manifolds, ZAMM Z. Angew. Math. Mech., 93 (2013), 446-458. doi: 10.1002/zamm.201100194.  Google Scholar [27] M. Kohr and I. Pop, "Viscous Incompressible Flow for Low Reynolds Numbers," WIT Press, Southampton (UK), 2004.  Google Scholar [28] 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), 217-235.  Google Scholar [29] V. Maz'ya and J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Ration. Mech. Anal., 194 (2009), 669?12. doi: 10.1007/s00205-008-0171-z.  Google Scholar [30] D. Medková, Transmission problem for the Laplace equation and the integral equation method, J. Math. Anal. Appl., 387 (2012), 837-843. doi: 10.1016/j.jmaa.2011.09.041.  Google Scholar [31] O. Mendez and M. Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl., 6 (2000), 503-531.  Google Scholar [32] 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), 361-397.  Google Scholar [33] D. Mitrea, M. Mitrea and M. Taylor, Layer Potentials, the Hodge Laplacian and Global Boundary Problems in Non-Smooth Riemannian Manifolds, Memoirs Amer. Math. Soc., 150 (2001), No. 713. doi: 10.1090/memo/0713.  Google Scholar [34] M. Mitrea, S. Monniaux and M. Wright, , The Stokes operator with Neumann boundary conditions in Lipschitz domains, J. Math. Sci. (New York), 176 (2011), 409-457.  Google Scholar [35] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal., 163 (1999), 181-251.  Google Scholar [36] 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), 1-79. doi: 10.1006/jfan.2000.3619.  Google Scholar [37] M. Mitrea and M. Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann., 321 (2001), 955-987. doi: 10.1007/s002080100261.  Google Scholar [38] M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, 344 (2012), viii+241 pp.  Google Scholar [39] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar [40] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Publ. Co. Amsterdam, 1978.  Google Scholar [41] J. T. Wloka, B. Rowley and B. Lawruk, "Boundary Value Problems for Elliptic Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar

show all references

##### References:
 [1] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glasnik Matematički, 35 (2000), 161-177.  Google Scholar [2] C. Băcuţă, A. L. Mazzucato, V. Nistor and L. Zikatanov, Interface and mixed boundary value problems on n-dimensional polyhedral domains, Documenta Math., 15 (2010), 687-745.  Google Scholar [3] J. K. Choi and S. Kim, Neumann functions for second order elliptic systems with measurable coefficients, Trans. Amer. Math. Soc., 2013, to appear. doi: 10.1090/S0002-9947-2013-05886-2.  Google Scholar [4] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal., 19 (1988), 613-626. doi: 10.1137/0519043.  Google Scholar [5] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J., 65 (1992), 333-343. doi: 10.1215/S0012-7094-92-06514-8.  Google Scholar [6] B. E. J. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math., 125 (1987), 437-465.  Google Scholar [7] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari (Italian), Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43.  Google Scholar [8] 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), 1-47. doi: 10.1007/s00205-004-0320-y.  Google Scholar [9] D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-163.  Google Scholar [10] L. Escauriaza and M. Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains, J. Funct. Anal., 216 (2004), 141-171. doi: 10.1016/j.jfa.2003.12.005.  Google Scholar [11] E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J., 57 (1988), 769-793. doi: 10.1215/S0012-7094-88-05734-1.  Google Scholar [12] D. Fericean, Layer potential analysis of a Neumann problem for the Brinkman system,, Mathematica (Cluj), ().   Google Scholar [13] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 Edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar [14] S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172. doi: 10.1007/s00229-007-0107-1.  Google Scholar [15] S. Hofmann, M. Mitrea and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, Int. Math. Res. Notices, 14 (2010), 2567-2865. doi: 10.1093/imrn/rnp214.  Google Scholar [16] G. C. Hsiao and W. L. Wendland, "Boundary Integral Equations," Springer, Heidelberg, 2008.  Google Scholar [17] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accesible domains, Adv. Math., 46 (1982), 80-147. doi: 10.1016/0001-8708(82)90055-X.  Google Scholar [18] N. J. Kalton, S. Mayboroda and M. Mitrea, Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations, Contemp. Math., 445 (2007), 121-177.  Google Scholar [19] N. J. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc., 350 (1998), 3903-3922.  Google Scholar [20] K. Kang and S. Kim, Global pointwise estimates for Green's matrix of second order elliptic systems,, arXiv:1001.2618v2, ().   Google Scholar [21] M. Kohr, M. Lanza de Cristoforis and W. L. Wendland, Nonlinear Neumann-transmission problems for Stokes and Brinkman equations on Euclidean Lipschitz domains, Potential Anal., 38 (2013), 1123-1171. doi: 10.1007/s11118-012-9310-0.  Google Scholar [22] 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), 493-537. doi: 10.3934/cpaa.2010.9.493.  Google Scholar [23] 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), 229-273. doi: 10.1007/s11118-009-9151-7.  Google Scholar [24] M. Kohr, C. Pintea and W. L. Wendland, Dirichlet-transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 999-1018. doi: 10.3934/dcdsb.2011.15.999.  Google Scholar [25] M. Kohr, C. Pintea and W. L. Wendland, Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: Applications to pseudodifferential Brinkman operators, Int. Math. Res. Notices, 2012, DOI 10.1093/imrn/RNS158, to appear. doi: 10.1093/imrn/rns158.  Google Scholar [26] M. Kohr, C. Pintea and W. L. Wendland, Dirichlet-transmission problems for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in Riemannian manifolds, ZAMM Z. Angew. Math. Mech., 93 (2013), 446-458. doi: 10.1002/zamm.201100194.  Google Scholar [27] M. Kohr and I. Pop, "Viscous Incompressible Flow for Low Reynolds Numbers," WIT Press, Southampton (UK), 2004.  Google Scholar [28] 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), 217-235.  Google Scholar [29] V. Maz'ya and J. Rossmann, Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains, Arch. Ration. Mech. Anal., 194 (2009), 669?12. doi: 10.1007/s00205-008-0171-z.  Google Scholar [30] D. Medková, Transmission problem for the Laplace equation and the integral equation method, J. Math. Anal. Appl., 387 (2012), 837-843. doi: 10.1016/j.jmaa.2011.09.041.  Google Scholar [31] O. Mendez and M. Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl., 6 (2000), 503-531.  Google Scholar [32] 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), 361-397.  Google Scholar [33] D. Mitrea, M. Mitrea and M. Taylor, Layer Potentials, the Hodge Laplacian and Global Boundary Problems in Non-Smooth Riemannian Manifolds, Memoirs Amer. Math. Soc., 150 (2001), No. 713. doi: 10.1090/memo/0713.  Google Scholar [34] M. Mitrea, S. Monniaux and M. Wright, , The Stokes operator with Neumann boundary conditions in Lipschitz domains, J. Math. Sci. (New York), 176 (2011), 409-457.  Google Scholar [35] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal., 163 (1999), 181-251.  Google Scholar [36] 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), 1-79. doi: 10.1006/jfan.2000.3619.  Google Scholar [37] M. Mitrea and M. Taylor, Navier-Stokes equations on Lipschitz domains in Riemannian manifolds, Math. Ann., 321 (2001), 955-987. doi: 10.1007/s002080100261.  Google Scholar [38] M. Mitrea and M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, 344 (2012), viii+241 pp.  Google Scholar [39] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar [40] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Publ. Co. Amsterdam, 1978.  Google Scholar [41] J. T. Wloka, B. Rowley and B. Lawruk, "Boundary Value Problems for Elliptic Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar
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