Abstract
We derive an \(H_{0}^{s,p}(\text {curl};\varOmega )\) estimate for the solutions of the Maxwell type equations modeled with anisotropic and \(W^{s, \infty }(\varOmega )\)-regular coefficients. Here, we obtain the regularity of the solutions for the integrability and smoothness indices \((p, s)\) in a plane domain characterized by the apriori lower/upper bounds of \(a\) and the apriori upper bound of its Hölder semi-norm of order \(s\). The proof relies on a perturbation argument generalizing Gröger’s \(L^p\)-type estimate, known for the elliptic problems, to the Maxwell system.
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Notes
One way to deal with the general case where \(k^2\) is replaced by \(b \in (L^{\infty }(\varOmega ))^{3\times 3}\) (for example in (1.4) replace \(k^2\) by \(b(x):=-k^2 \epsilon ^{-1}(x)\), \(x\in \varOmega \), with \(\epsilon \in (L^\infty (\varOmega ))^{3\times 3}\) lower bounded by a positive constant) is discussed in Remark 1.
The space \(L_{s}^{p}({\mathbb {R}}^3)\) can also be defined using the Fourier transform \( L_{s}^{p}({\mathbb {R}}^3) := \{f ; f\in {\mathcal {S}}', \Vert f\Vert _{p}^{s} < \infty \}, \) with the norm \( \Vert f\Vert _{p}^{s} = \Vert {\mathcal {F}}^{-1}\{(1+|\xi |^2)^{\frac{s}{2}}{\mathcal {F}}{f}\} \Vert _{L^p({\mathbb {R}}^3)}, \) where \(s\in {\mathbb {R}}\). Here, \({\mathcal {F}}\) and \({\mathcal {F}}^{-1}\) represent the Fourier transform and inverse Fourier transform respectively and \({\mathcal {S}}'\) represents the space of tempered distributions.
In the case \(s=0\), this condition is not needed. In all subsequent estimates, we can replace \(\tilde{M}\) by \(0\) in this case.
References
Adams, R.A.: Sobolev spaces. Pure and Applied Mathematics, vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975)
Alberti, G.S., Capdeboscq, Y.: Elliptic regularity theory applied to time harmonic anisotropic Maxwell’s equations with less than Lipschitz complex coefficients. SIAM J. Math. Anal. 46(1), 998–1016 (2014)
Bao, G., Li, Y., Zhou, Z.: \(L^p\) estimates of time-harmonic Maxwell’s equations in a bounded domain. J. Differ. Equ. 245(12), 3674–3686 (2008)
Bao, G., Minut, A., Zhou, Z.: \(L^p\) estimates for Maxwell’s equations with source term. Commun. Partial Differ. Equ. 32(7–9), 1449–1471 (2007)
Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin (1976). Grundlehren der Mathematischen Wissenschaften, No. 223
Evans, L.C.: Partial differential equations, volume 19 of Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition
Grafakos, L., Oh, S.: The Kato-Ponce inequality. Commun. Partial Differ. Equ. 39(6), 1128–1157 (2014)
Gröger, K.: A \(W^{1, p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283(4), 679–687 (1989)
Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)
Kar, M., Sini, M.: Reconstruction of interfaces using CGO solutions for the Maxwell equations. J. Inverse Ill-Posed Probl. 22(2), 169–208 (2014)
Meyers, N.G.: An \(L^{p}\)-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 3(17), 189–206 (1963)
Mitrea, D., Mitrea, M., Pipher, J.: Vector potential theory on nonsmooth domains in \(\mathbf{R}^3\) and applications to electromagnetic scattering. J. Fourier Anal. Appl. 3(2), 131–192 (1997)
Mitrea, M.: Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds. Duke Math. J. 125(3), 467–547 (2004)
Monk, P.: Finite element methods for Maxwell’s equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003)
Rudin, W.: Functional analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)
Sini, M., Yoshida, K.: On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case. Inverse Probl. 28(5), 055013 (2012). 22
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)
Triebel, H.: Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers. Rev. Mat. Complut. 15(2), 475–524 (2002)
Yin, H.M.: Regularity of weak solution to Maxwell’s equations and applications to microwave heating. J. Differ. Equ. 200(1), 137–161 (2004)
Acknowledgments
MK was supported by the Academy of Finland through the Finnish Centre of Excellence in Inverse Problems Research and the ERC Starting Grant (Grant Agreement No 307023) and he is very thankful to Mikko Salo for his support. MS was partially supported by RICAM. The authors also would like to express their gratitude to the University of Jyväskylä, Finland and RICAM, Austrian Academy of Sciences, Austria, where most of the work has been done.
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Kar, M., Sini, M. An \(H^{s,p}(\text {curl};\varOmega )\) estimate for the Maxwell system. Math. Ann. 364, 559–587 (2016). https://doi.org/10.1007/s00208-015-1225-9
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DOI: https://doi.org/10.1007/s00208-015-1225-9