Time-harmonic Maxwell equations with asymptotically linear polarization

Qin, Dongdong; Tang, Xianhua

doi:10.1007/s00033-016-0626-2

Cited by 11 publications

(22 citation statements)

References 35 publications

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“…Now we discuss asymptotically linear nonlinearities. So far these have only been considered in [38]. Here we present a variation of their main result [38, Theorem 1.1] within our setting.…”

Section: The Bounded Domain Casementioning

confidence: 91%

“…If N = ∅ one can still obtain a critical point of J via minimization over N . This has been done in [38].…”

Section: Critical Point Theorymentioning

confidence: 99%

“…In this section we consider the curl-curl equation (1.4) on a bounded Lipschitz domain Ω, and we present results from [11,12,38,50]. Recall the hypotheses (L1), (F1)-(F3) from Section 2 which imply that the functional…”

Section: The Bounded Domain Casementioning

confidence: 99%

“…Here we present a variation of their main result [38, Theorem 1.1] within our setting. The main difference is that V (x) ≡ λ > 0 is a constant scalar in [38].…”

Section: The Bounded Domain Casementioning

confidence: 99%

“…Then in Section 3 we present a theorem in a simpler symmetric setting where standard methods from critical point theory can be applied, which can be found in the seminal paper [1] by Ambrosetti and Rabinowitz and the book [39] by Rabinowitz. In order to treat the full strongly indefinite functional we recall some critical point theory developed for (1.4). Then in Section 5 we discuss results from [11,12,38,50] about (1.4) on bounded domains. The case Ω = R 3 will be discussed in Section 6.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear time-harmonic Maxwell equations in domains

Bartsch

Mederski

2017

J. Fixed Point Theory Appl.

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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

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“…Now we discuss asymptotically linear nonlinearities. So far these have only been considered in [38]. Here we present a variation of their main result [38, Theorem 1.1] within our setting.…”

Section: The Bounded Domain Casementioning

confidence: 91%

“…If N = ∅ one can still obtain a critical point of J via minimization over N . This has been done in [38].…”

Section: Critical Point Theorymentioning

confidence: 99%

Section: The Bounded Domain Casementioning

confidence: 99%

“…Here we present a variation of their main result [38, Theorem 1.1] within our setting. The main difference is that V (x) ≡ λ > 0 is a constant scalar in [38].…”

Section: The Bounded Domain Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear time-harmonic Maxwell equations in domains

Bartsch

Mederski

2017

J. Fixed Point Theory Appl.

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Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

Qin

Tang

2017

Math Methods in App Sciences

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This paper is focused on following time‐harmonic Maxwell equation: ∇×false(μ−1false(xfalse)∇×ufalse)−ω2εfalse(xfalse)u=ffalse(x,ufalse),2emin.4emnormalΩ,ν×u=0,2em2emon.4em∂normalΩ, where normalΩ⊂R3 is a bounded Lipschitz domain, ν:∂normalΩ→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as false|ufalse|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor ε∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

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Vortex Ground State Solutions for Electromagnetostatic Schrödinger–Maxwell System with Critical Exponent

Ji,

Teng

2024

Mediterr. J. Math.

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Time-harmonic Maxwell equations with asymptotically linear polarization

Cited by 11 publications

References 35 publications

Nonlinear time-harmonic Maxwell equations in domains

Nonlinear time-harmonic Maxwell equations in domains

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

Vortex Ground State Solutions for Electromagnetostatic Schrödinger–Maxwell System with Critical Exponent

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