June  2015, 8(3): 475-496. doi: 10.3934/dcdss.2015.8.475

Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received  October 2013 Revised  June 2014 Published  October 2014

We show that $L^p$ vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We use these tools to investigate the regularity of the solution to the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation `in H' of the problem, in a reference geometrical setting allowing for material heterogeneities and nonsmooth interfaces.
Citation: Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475
References:
[1]

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), 823-864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[2]

A. Bossavit, Electromagnetism in View of Modeling, Springer, Berlin, Heidelberg, New York, 2004, (French). Google Scholar

[3]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368. doi: 10.1002/mma.1670120406.  Google Scholar

[4]

M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equat. Oper. Th., 15 (1992), 227-261. doi: 10.1007/BF01204238.  Google Scholar

[5]

W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys., 15 (2013), 7075-7086. doi: 10.1039/c3cp44390f.  Google Scholar

[6]

P.-E. Druet, Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries, Preprint 1270 of the Weierstrass Institute for Applied mathematics and Stochastics, Berlin, 2007, Available in pdf-format at http://www.wias-berlin.de/preprint/1270/wias_preprints_1270.pdf. Google Scholar

[7]

P.-E. Druet, Analysis of a Coupled System of Partial Differential Equations Modeling the Interaction Between Melt Flow, Global Heat Transfer and Applied Magnetic Fields in Crystal Growth, PhD thesis, Humboldt Universität zu Berlin, Germany, 2009, Available at http://edoc.hu-berlin.de/dissertationen. Google Scholar

[8]

P.-E. Druet, W. Dreyer, O. Klein and J. Sprekels, Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals, Milan J. Math., 80 (2012), 311-332. doi: 10.1007/s00032-012-0184-9.  Google Scholar

[9]

P.-E. Druet, O. Klein, J. Sprekels, F. Tröltzsch and J. Yousept, Optimal control of 3D state-constrained induction heating processes, SIAM J. Control Optim., 49 (2011), 1707-1736. doi: 10.1137/090760544.  Google Scholar

[10]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.  Google Scholar

[11]

J. Elschner, J. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces Free Bound., 9 (2007), 233-252. doi: 10.4171/IFB/163.  Google Scholar

[12]

I. Gasser and P. Marcati, On a generalization of the DIV-CURL lemma, Osaka J. Math., 45 (2008), 211-214.  Google Scholar

[13]

R. Griesinger, The boundary value problem rot $u = f$, $u$ vanishing at the boundary and the related decomposition of $L^q$ and $H^{1,q}_0$, Ann. Univ. Ferrara - Sez. VII - Sc. Mat., 26 (1990), 15-43.  Google Scholar

[14]

R. Haller-Dintelmann, H.-C. Kaiser, J. Rehberg and G. Schmidt, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de mathématique pures et appliquées, 89 (2008), 25-48. doi: 10.1016/j.matpur.2007.09.001.  Google Scholar

[15]

D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control of the thermistor problem, SIAM J. Control Optim., 48 (2010), 3449-3481. doi: 10.1137/080736259.  Google Scholar

[16]

D. Hömberg and E. Rocca, A model for resistance welding including phase transitions and Joule heating, Math. Methods Appl. Sci., 34 (2011), 2077-2088. doi: 10.1002/mma.1505.  Google Scholar

[17]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Functional Analysis, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.  Google Scholar

[18]

F. Jochmann, A compactness result for vector fields with divergence and curl in $L^q(\Omega)$ involving mixed boundary conditions, Applicable Analysis, 66 (1997), 189-203. doi: 10.1080/00036819708840581.  Google Scholar

[19]

O. Klein, P. Philip and J. Sprekels, Modelling and simulation of sublimation growth in SiC bulk single crystals, Interfaces and Free Boundaries, 6 (2004), 295-314. doi: 10.4171/IFB/101.  Google Scholar

[20]

A. Kufner, O. John and S. Fučik, Function Spaces, Academia Prague, Prague, 1977.  Google Scholar

[21]

O. Ladyzhenskaja and V. Solonnikov, Solutions of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid, Trudy Mat. Inst. Steklov, 59 (1960), 115-173, Russian.  Google Scholar

[22]

N. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206.  Google Scholar

[23]

P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar

[24]

R. Picard, An elementary proof for a compact embedding result in generalized electromagnetic theory, Math. Z., 187 (1984), 151-164. doi: 10.1007/BF01161700.  Google Scholar

[25]

R. Picard, On the low-frequency asymptotics in electromagnetic theory, J. Reine Angew. Math., 354 (1984), 50-73. doi: 10.1515/crll.1984.354.50.  Google Scholar

[26]

R. Picard and A. Milani, Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems, in Partial differential equations and calculus of variations, 1357 of Lect. Notes Math., Springer, 1999, 317-340. doi: 10.1007/BFb0082873.  Google Scholar

[27]

J. Robbin, R. Rogers and B. Temple, On weak continuity and the Hodge decomposition, Trans. Amer. Math. Soc., 303 (1987), 609-618. doi: 10.1090/S0002-9947-1987-0902788-8.  Google Scholar

[28]

W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$., Math. Meth. Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.  Google Scholar

[29]

D. D. Zanger, The inhomogeneous Neumann problem in Lipschitz domains, Commun. in Partial Differential Equations, 25 (2000), 1771-1808. doi: 10.1080/03605302.2000.10824220.  Google Scholar

show all references

References:
[1]

C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), 823-864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[2]

A. Bossavit, Electromagnetism in View of Modeling, Springer, Berlin, Heidelberg, New York, 2004, (French). Google Scholar

[3]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368. doi: 10.1002/mma.1670120406.  Google Scholar

[4]

M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equat. Oper. Th., 15 (1992), 227-261. doi: 10.1007/BF01204238.  Google Scholar

[5]

W. Dreyer, C. Guhlke and R. Müller, Overcoming the shortcomings of the Nernst-Planck model, Phys. Chem. Chem. Phys., 15 (2013), 7075-7086. doi: 10.1039/c3cp44390f.  Google Scholar

[6]

P.-E. Druet, Higher integrability of the Lorentz force for weak solutions to Maxwell's equations in complex geometries, Preprint 1270 of the Weierstrass Institute for Applied mathematics and Stochastics, Berlin, 2007, Available in pdf-format at http://www.wias-berlin.de/preprint/1270/wias_preprints_1270.pdf. Google Scholar

[7]

P.-E. Druet, Analysis of a Coupled System of Partial Differential Equations Modeling the Interaction Between Melt Flow, Global Heat Transfer and Applied Magnetic Fields in Crystal Growth, PhD thesis, Humboldt Universität zu Berlin, Germany, 2009, Available at http://edoc.hu-berlin.de/dissertationen. Google Scholar

[8]

P.-E. Druet, W. Dreyer, O. Klein and J. Sprekels, Mathematical modeling of Czochralski type growth processes for semiconductor bulk single crystals, Milan J. Math., 80 (2012), 311-332. doi: 10.1007/s00032-012-0184-9.  Google Scholar

[9]

P.-E. Druet, O. Klein, J. Sprekels, F. Tröltzsch and J. Yousept, Optimal control of 3D state-constrained induction heating processes, SIAM J. Control Optim., 49 (2011), 1707-1736. doi: 10.1137/090760544.  Google Scholar

[10]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.  Google Scholar

[11]

J. Elschner, J. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces Free Bound., 9 (2007), 233-252. doi: 10.4171/IFB/163.  Google Scholar

[12]

I. Gasser and P. Marcati, On a generalization of the DIV-CURL lemma, Osaka J. Math., 45 (2008), 211-214.  Google Scholar

[13]

R. Griesinger, The boundary value problem rot $u = f$, $u$ vanishing at the boundary and the related decomposition of $L^q$ and $H^{1,q}_0$, Ann. Univ. Ferrara - Sez. VII - Sc. Mat., 26 (1990), 15-43.  Google Scholar

[14]

R. Haller-Dintelmann, H.-C. Kaiser, J. Rehberg and G. Schmidt, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de mathématique pures et appliquées, 89 (2008), 25-48. doi: 10.1016/j.matpur.2007.09.001.  Google Scholar

[15]

D. Hömberg, C. Meyer, J. Rehberg and W. Ring, Optimal control of the thermistor problem, SIAM J. Control Optim., 48 (2010), 3449-3481. doi: 10.1137/080736259.  Google Scholar

[16]

D. Hömberg and E. Rocca, A model for resistance welding including phase transitions and Joule heating, Math. Methods Appl. Sci., 34 (2011), 2077-2088. doi: 10.1002/mma.1505.  Google Scholar

[17]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Functional Analysis, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067.  Google Scholar

[18]

F. Jochmann, A compactness result for vector fields with divergence and curl in $L^q(\Omega)$ involving mixed boundary conditions, Applicable Analysis, 66 (1997), 189-203. doi: 10.1080/00036819708840581.  Google Scholar

[19]

O. Klein, P. Philip and J. Sprekels, Modelling and simulation of sublimation growth in SiC bulk single crystals, Interfaces and Free Boundaries, 6 (2004), 295-314. doi: 10.4171/IFB/101.  Google Scholar

[20]

A. Kufner, O. John and S. Fučik, Function Spaces, Academia Prague, Prague, 1977.  Google Scholar

[21]

O. Ladyzhenskaja and V. Solonnikov, Solutions of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid, Trudy Mat. Inst. Steklov, 59 (1960), 115-173, Russian.  Google Scholar

[22]

N. Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 189-206.  Google Scholar

[23]

P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar

[24]

R. Picard, An elementary proof for a compact embedding result in generalized electromagnetic theory, Math. Z., 187 (1984), 151-164. doi: 10.1007/BF01161700.  Google Scholar

[25]

R. Picard, On the low-frequency asymptotics in electromagnetic theory, J. Reine Angew. Math., 354 (1984), 50-73. doi: 10.1515/crll.1984.354.50.  Google Scholar

[26]

R. Picard and A. Milani, Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems, in Partial differential equations and calculus of variations, 1357 of Lect. Notes Math., Springer, 1999, 317-340. doi: 10.1007/BFb0082873.  Google Scholar

[27]

J. Robbin, R. Rogers and B. Temple, On weak continuity and the Hodge decomposition, Trans. Amer. Math. Soc., 303 (1987), 609-618. doi: 10.1090/S0002-9947-1987-0902788-8.  Google Scholar

[28]

W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$., Math. Meth. Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.  Google Scholar

[29]

D. D. Zanger, The inhomogeneous Neumann problem in Lipschitz domains, Commun. in Partial Differential Equations, 25 (2000), 1771-1808. doi: 10.1080/03605302.2000.10824220.  Google Scholar

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