Far field refraction problem with loss of energy in negative refractive index material

Haokun Sui; Feida Jiang; Haokun Sui; Feida Jiang

doi:10.3934/mine.2026004

Mathematics in Engineering

2026, Volume 8, Issue 1: 98-139. doi: 10.3934/mine.2026004

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Research article

Far field refraction problem with loss of energy in negative refractive index material

Haokun Sui ¹,
Feida Jiang ^{1,2
,
,}

1.
School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University, Nanjing 211189, China
2.
Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, China

Received: 30 September 2025 Revised: 26 January 2026 Accepted: 28 January 2026 Published: 02 February 2026

This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index $ \kappa $, that is, $ \kappa < -1 $ and $ -1 < \kappa < 0 $. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Ampère type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.
- negative refraction,
- far field,
- weak solution,
- loss of energy,
- Monge-Ampère type operator
Citation: Haokun Sui, Feida Jiang. Far field refraction problem with loss of energy in negative refractive index material[J]. Mathematics in Engineering, 2026, 8(1): 98-139. doi: 10.3934/mine.2026004

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Abstract

This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index $ \kappa $, that is, $ \kappa < -1 $ and $ -1 < \kappa < 0 $. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Ampère type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.

References

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