# 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 & 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.  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, (2004).   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.  doi: 10.1002/mma.1670120406.  Google Scholar [4] M. Dauge, Neumann and mixed problems on curvilinear polyhedra,, Integr. Equat. Oper. Th., 15 (1992), 227.  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.  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, (1270).   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, (2009).   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.  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.  doi: 10.1137/090760544.  Google Scholar [10] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer, (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.  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.   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.   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.  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.  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.  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.  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.  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.  doi: 10.4171/IFB/101.  Google Scholar [20] A. Kufner, O. John and S. Fučik, Function Spaces,, Academia 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.   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.   Google Scholar [23] P. Monk, Finite Element Methods for Maxwell's Equations,, Clarendon press, (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.  doi: 10.1007/BF01161700.  Google Scholar [25] R. Picard, On the low-frequency asymptotics in electromagnetic theory,, J. Reine Angew. Math., 354 (1984), 50.  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 (1999), 317.  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.  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.  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.  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.  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, (2004).   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.  doi: 10.1002/mma.1670120406.  Google Scholar [4] M. Dauge, Neumann and mixed problems on curvilinear polyhedra,, Integr. Equat. Oper. Th., 15 (1992), 227.  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.  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, (1270).   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, (2009).   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.  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.  doi: 10.1137/090760544.  Google Scholar [10] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer, (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.  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.   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.   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.  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.  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.  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.  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.  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.  doi: 10.4171/IFB/101.  Google Scholar [20] A. Kufner, O. John and S. Fučik, Function Spaces,, Academia 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.   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.   Google Scholar [23] P. Monk, Finite Element Methods for Maxwell's Equations,, Clarendon press, (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.  doi: 10.1007/BF01161700.  Google Scholar [25] R. Picard, On the low-frequency asymptotics in electromagnetic theory,, J. Reine Angew. Math., 354 (1984), 50.  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 (1999), 317.  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.  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.  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.  doi: 10.1080/03605302.2000.10824220.  Google Scholar
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