Nonlinear time-harmonic Maxwell equations in domains

Abstract

The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equationfor the field u : Ω → R 3 in a domain Ω ⊂ R 3 . Here ε(x) ∈ R 3×3 is the (linear) permittivity tensor of the material, and µ(x) ∈ R 3×3 denotes the magnetic permeability tensor. The nonlinearity f : Ω × R 3 → R 3 comes from the nonlinear polarization. If f = ∇ u F 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. MSC 2010: Primary: 35Q60; Secondary: 35J20, 58E05, 78A25 for the electric field E. Solving this one obtains D = εE + P N L (x, E) by the constitutive relation and B by time integrating Faraday's law. Finally H = µ −1 B is also determined by the constitutive relation. The fields B and D will automatically be divergence free provided they are divergence free at time 0.The field E is said to be time-harmonic (also monochromatic) with frequency ω > 0 if E(x, t) = E 1 (x) cos(ωt) + E 2 (x) sin(ωt) for x ∈ Ω and t ∈ R.The intensity of a time-harmonic field E is defined as the time-average 1 T T 0 |E(x, t)| 2 dt = |E 1 (x)| 2 + |E 2 (x)| 2

Multiple Solutions to a Nonlinear Curl–Curl Problem in $${\mathbb {R}}^3$$

“…Similar difficulties have already appeared in curl-curl problems on bounded domains in Bartsch and Mederski [4] where a generalized Nehari manifold approach inspired by Szulkin and Weth [36] has been developed to overcome strong indefiniteness. Other approaches have been developed in subsequent work [5,27]; see also the survey [6]. Note that on a bounded domain there is no problem with lack of weak-to-weak * continuity of E ′ since a variant of the Palais-Smale condition is satisfied under some constraints.…”

Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium

“…Nonlinearities of the form (2.3) have not been dealt with in [6] because it was unclear whether they satisfy the hypothesis (*) from Remark 2.1 d). Given the other conditions from Theorem 2.2, it has been observed in [7,Remark 5.4 (d)] that a weaker variant of (*) is essentially equivalent to (F9) from [7], which is a stronger variant of (F5). Now we concentrate on nonlinear uniaxial media which are of great importance due to the phenomenon of birefringence and applications in crystallography [25,27,31].…”

Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations

“…where s ∈ [1, ∞] and p as in (4). We stress that only conditions near zero are needed since we are going to construct small solutions in L q (R n ) which will turn out to be small also in L ∞ (R n ).…”