# American Institute of Mathematical Sciences

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 and 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. [2] A. Bossavit, Electromagnetism in View of Modeling, Springer, Berlin, Heidelberg, New York, 2004, (French). [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. [4] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equat. Oper. Th., 15 (1992), 227-261. doi: 10.1007/BF01204238. [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. [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. [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. [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. [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. [10] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. [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. [12] I. Gasser and P. Marcati, On a generalization of the DIV-CURL lemma, Osaka J. Math., 45 (2008), 211-214. [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. [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. [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. [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. [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. [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. [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. [20] A. Kufner, O. John and S. Fučik, Function Spaces, Academia Prague, Prague, 1977. [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. [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. [23] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. [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. [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. [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. [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. [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. [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.

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##### 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. [2] A. Bossavit, Electromagnetism in View of Modeling, Springer, Berlin, Heidelberg, New York, 2004, (French). [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. [4] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equat. Oper. Th., 15 (1992), 227-261. doi: 10.1007/BF01204238. [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. [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. [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. [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. [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. [10] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. [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. [12] I. Gasser and P. Marcati, On a generalization of the DIV-CURL lemma, Osaka J. Math., 45 (2008), 211-214. [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. [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. [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. [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. [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. [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. [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. [20] A. Kufner, O. John and S. Fučik, Function Spaces, Academia Prague, Prague, 1977. [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. [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. [23] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001. [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. [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. [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. [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. [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. [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.
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