Singularities of Maxwell's system in non-hilbertian Sobolev spaces

Chikouche, Wided; Nicaise, Serge

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 3, pp. 455-482.

We study the regularity of the solution of the regularized electric Maxwell problem in a polygonal domain with data in $L^p{\left(\Omega \right)}^2$. Using a duality method, we prove a decomposition of the solution into a regular part in the non-Hilbertian Sobolev space $W^{2,p}{\left(\Omega \right)}^2$ and an explicit singular one.

Classification : 35A20, 35Q60, 78A25

[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general conditions II, Comm. Pure Appl. Math. 17 (1964), 35-92. | MR | Zbl

[2] A. Aibeche, W. Chikouche and S. Nicaise, $l^p$ regularity of transmission problems in dihedral domains, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10-B (2007), 663-660. | MR | Zbl

[3] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Methods Appl. Sci. 21 (1998), 823-864. | MR | Zbl

[4] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci. 21 (1998), 519-549. | MR | Zbl

[5] F. Assous and P. Ciarlet, Une caractérisation de l’orthogonal de $\Delta (H^2\left(\Omega \right)\cap H_0^1\left(\Omega \right))$ dans $L^2\left(\Omega \right)$, C. R. Acad. Sc. Paris, Série I 325 (1997), 605-610. | MR | Zbl

[6] F. Assous, P. Ciarlet Jr. and E. Sonnendrücker, Résolution of the Maxwell equations in a domain with reentrant corners, RAIRO Modél. Math. Anal. Numér. 32 (1998), 359-389. | Numdam | MR | Zbl

[7] M. Birman and M. Z. Solomyak, $L^2$ theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys 42 (1987), 75-96. | MR | Zbl

[8] M. S. Birman and M. Z. Solomyak, On the main singularities of the electric component of the electro-magnetic field in regions with screens, St. Petersburg Math. J. 5 (1993), 125-139. | MR | Zbl

[9] P. Clément and P. Grisvard, Somme d’opérateurs et régularité $L^p$ dans les problèmes aux limites, C. R. Acad. Sci. Paris Série I, 314 (1992), 821-824. | MR | Zbl

[10] M. Costabel A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci. 12 (1990), 365-368. | MR | Zbl

[11] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains Arch. Ration. Mech. Anal. 151 (2000), 221-276. | MR | Zbl

[12] G. Da Prato and P. Grisvard, Somme d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., Ser. IX, 54 (1975), 305-387. | MR | Zbl

[13] G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201. | MR | Zbl

[14] V. Giraultand and P.-A. Raviart, “Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms”, Vol. 5, Springer Series in Computational Mathematics, Springer, Berlin, 1986. | Zbl

[15] P. Grisvard, “Elliptic Problems in Nonsmooth Domains”, Vol. 24, Monographs and Studies in Mathematics, Pitman, Boston-London-Melbourne, 1985. | MR | Zbl

[16] M. D. Gunzburger, A. J. Meir and J. S. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary,incompressible magnetohydrodynamics, Math. Comp. 56 (1991), 523-563. | MR | Zbl

[17] V. G. Maz'Ya and B. A. Plamenevskii, Estimates in $L^p$ and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Amer. Math. Soc. Transl. Ser. 2, 123 (1984), 1-56. | Zbl

[18] M. Moussaoui Espace $H(\mathrm{curl},div)$ dans un polygone plan, C. R. Acad. Sc. Paris, Série I, 322 (1996), 225-229. | Zbl

[19] S. Nicaise, “ Polygonal Interface Problems”, Vol. 39, Methoden und Verfahren der mathematischen Physik, Peter Lang GmbH, Europäischer Verlag der Wissenschaften, Frankfurt/M., 1993. | MR | Zbl

[20] S. Nicaise, Edge elements on anisotropic meshes and approximation of the Maxwell equations, SIAM J. Numer. Anal. 39 (2001), 784-816(electronic). | MR | Zbl

[21] C. Weber A local compactness theorem for Maxwell's equations, Math. Methods Appl. Sci. 2 (1980), 12-25. | MR | Zbl