Abstract
The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation
for the field \(u:\Omega \rightarrow \mathbb {R}^3\) in a domain \(\Omega \subset \mathbb {R}^3\). Here, \(\varepsilon (x) \in \mathbb {R}^{3\times 3}\) is the (linear) permittivity tensor of the material, and \(\mu (x) \in \mathbb {R}^{3\times 3}\) denotes the magnetic permeability tensor. The nonlinearity \(f:\Omega \times \mathbb {R}^3\rightarrow \mathbb {R}^3\) comes from the nonlinear polarization. If \(f=\nabla _uF\) is a gradient, then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.
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References
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21(9), 823–864 (1998)
Azzollini, A., Benci, V., D’Aprile, T., Fortunato, D.: Existence of static solutions of the semilinear Maxwell equations. Ric. Mat. 55(2), 283–297 (2006)
Badiale, M., Pisani, L., Rolando, S.: Sum of weighted Lebesgue spaces and nonlinear elliptic equations. Nonlinear Differ. Equ. Appl. 18, 369–405 (2011)
Ball, J.M., Capdeboscq, Y., Tsering-Xiao, B.: On uniqueness for time harmonic anisotropic Maxwell’s equations with piecewise regular coefficients. Math. Models Methods Appl. Sci. 22(11), 1250036, 11 pp (2012)
Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. Theory Methods Appl. 7, 981–1012 (1983)
Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. Theory Methods Appl. 20(12), 1205–1216 (1993)
Bartsch, T., Dancer, N., Wang, Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37, 345–361 (2010)
Bartsch, T., Ding, Y.: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Mathematische Nachrichten 279(12), 1267–1288 (2006)
Bartsch, T., Dohnal, T., Plum, M., Reichel, W.: Ground states of a nonlinear curl-curl problem in cylindrically symmetric media. Nonlinear Differ. Equ. Appl. 23:52(5), 34 pp. (2016)
Bartsch, T., Mederski, J.: Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain. Arch. Ration. Mech. Anal. 215(1), 283–306 (2015)
Bartsch, T., Mederski, J.: Nonlinear time-harmonic Maxwell equations in an anisotropic bounded domain. arXiv:1509.01994
Bauer, S., Pauly, D., Schomburg, M.: The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions. SIAM J. Math. Anal. 48(4), 2912–2943 (2016)
Benci, V., Fortunato, D.: Towards a unified field theory for classical electrodynamics. Arch. Ration. Mech. Anal. 173, 379–414 (2004)
Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. Math. 52(3), 241–273 (1979)
Buffa, A., Ammari, H., Nédélec, J.C.: Justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math. 60(5), 1805–1823 (2000)
Clark, D.C.: A variant of Lusternik–Schnirelmann theory. Indiana Univ. Math. J. 22, 65–74 (1972)
Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. Math. Model. Numer. Anal. 33, 627–649 (1999)
Costabel, M.: A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math. Methods Appl. Sci. 12, 365–368 (1990)
Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \({\mathbb{R}}^N\). Commun. Pure Appl. Math. 45(10), 1217–1269 (1992)
D’Aprile, T., Siciliano, G.: Magnetostatic solutions for a semilinear perturbation of the Maxwell equations. Adv. Differ. Equ. 16(5–6), 435–466 (2011)
Ding, Y.: Variational Methods for Strongly Indefinite Problems. Interdisciplinary Mathematical Sciences, vol. 7. World Scientific Publishing, Singapore (2007)
Dörfler, W., Lechleiter, A., Plum, M., Schneider, G., Wieners, C.: Photonic Crystals: Mathematical Analysis and Numerical Approximation. Springer, Basel (2012)
Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)
Hirsch, A., Reichel, W.: Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients. J. Anal. Appl. (to appear)
Kirsch, A., Hettlich, F.: The Mathematical Theory of Time-Harmonic Maxwell’s Equations: Expansion-, Integral-, and Variational Methods. Springer, Berlin (2015)
Leis, R.: Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien. Math. Z. 106, 213–224 (1968)
Mederski, J.: Ground states of time-harmonic semilinear Maxwell equations in \({\mathbb{R}}^3\) with vanishing permittivity. Arch. Ration. Mech. Anal. 218(2), 825–861 (2015)
Mederski, J.: Nonlinear time-harmonic Maxwell equations in \({\mathbb{R}}^3\): recent results and open questions. In: Lecture Notes of Seminario Interdisciplinare di Matematica, vol. 13, pp. 47–57 (2016)
Mederski, J.: Ground states of a system of nonlinear Schrödinger equations with periodic potentials. Commun. Partial Differ. Equ. 41(9), 1426–1440 (2016)
Mederski, J.: The Brezis–Nirenberg problem for the curl-curl operator. arXiv:1609.03989 (submitted)
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
Nie, W.: Optical nonlinearity: phenomena, applications, and materials. Adv. Mater. 5, 520–545 (1993)
Pankov, A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)
Picard, R., Weck, N., Witsch, K.-J.: Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles. Analysis (Munich) 21(3), 231–263 (2001)
Qin, D., Tang, X.: Time-harmonic Maxwell equations with asymptotically linear polarization. Z. Angew. Math. Phys. 67(3), 719–740 (2016)
Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986)
Pistoia, A., Ramos, M.: Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions. NoDEA Nonlinear Differ. Equ. Appl. 15(1), 1–23 (2008)
Saleh, B.E.A., Teich, M.C.: Fundamentals of Photonics, 2nd edn. Wiley, Hoboken (2007)
Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187(4), 511–517 (1984)
Stuart, C.A.: Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch. Ration. Mech. Anal. 113(1), 65–96 (1991)
Stuart, C.A.: Guidance properties of nonlinear planar waveguides. Arch. Ration. Mech. Anal. 125(1), 145–200 (1993)
Stuart, C.A., Zhou, H.S.: A variational problem related to self-trapping of an electromagnetic field. Math. Methods Appl. Sci. 19(17), 1397–1407 (1996)
Stuart, C.A., Zhou, H.S.: A constrained minimization problem and its application to guided cylindrical TM-modes in an anisotropic self-focusing dielectric. Calc. Var. Partial Differ. Equ. 16(4), 335–373 (2003)
Stuart, C.A., Zhou, H.S.: Axisymmetric TE-modes in a self-focusing dielectric. SIAM J. Math. Anal. 37(1), 218–237 (2005)
Stuart, C.A., Zhou, H.S.: Existence of guided cylindrical TM-modes in an inhomogeneous self-focusing dielectric. Math. Models Methods Appl. Sci. 20(9), 1681–1719 (2010)
Szulkin, A., Weth, T.: The method of Nehari Manifold. Handbook of Nonconvex Analysis and Applications, pp. 597–632. International Press, Somerville (2010)
Tang, X., Qin, D.: Ground state solutions for semilinear time-harmonic Maxwell equations. J. Math. Phys. 57(4), 041505 (2016)
Zeng, X.: Cylindrically symmetric ground state solutions for curl-curl equations with critical exponent. arXiv:1609.09598
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Dedicated to Paul Rabinowitz.
J. Mederski was partially supported by the National Science Centre, Poland (Grant No. 2014/15/D/ST1/03638).
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Bartsch, T., Mederski, J. Nonlinear time-harmonic Maxwell equations in domains. J. Fixed Point Theory Appl. 19, 959–986 (2017). https://doi.org/10.1007/s11784-017-0409-1
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DOI: https://doi.org/10.1007/s11784-017-0409-1
Keywords
- Time-harmonic Maxwell equations
- perfect conductor
- anisotropic media
- uniaxial media
- nonlinear material
- ground state
- variational methods
- strongly indefinite functional
- Nehari–Pankov manifold