American Institute of Mathematical Sciences

June  2011, 31(2): 525-543. doi: 10.3934/dcds.2011.31.525

Two dimensional invisibility cloaking via transformation optics

 1 Department of Mathematics and Statistics, University of Reading, Whiteknights, Reading RG6 6AX, United Kingdom 2 Department of Mathematics, University of California, Irvine, Irvine, CA 92697, United States

Received  June 2010 Revised  April 2011 Published  June 2011

We investigate two-dimensional invisibility cloaking via transformation optics approach. The cloaking media possess much more singular parameters than those having been considered for three-dimensional cloaking in literature. Finite energy solutions for these cloaking devices are studied in appropriate weighted Sobolev spaces. We derive some crucial properties of the singularly weighted Sobolev spaces. The invisibility cloaking is then justified by decoupling the underlying singular PDEs into one problem in the cloaked region and the other one in the cloaking layer. We derive some completely novel characterizations of the finite energy solutions corresponding to the singular cloaking problems. Particularly, some `hidden' boundary conditions on the cloaking interface are shown for the first time. We present our study for a very general system of PDEs, where the Helmholtz equation underlying acoustic cloaking is included as a special case.
Citation: Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 525-543. doi: 10.3934/dcds.2011.31.525
References:

show all references

References:
 [1] Guillermo Reyes, Juan-Luis Vázquez. The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1275-1294. doi: 10.3934/cpaa.2008.7.1275 [2] Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731 [3] Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005 [4] Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365 [5] Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885 [6] Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649 [7] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $C^{1}$ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [8] P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691 [9] Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427 [10] Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018 [11] Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565 [12] Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791 [13] Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [14] Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008 [15] Yutian Lei. On finite energy solutions of fractional order equations of the Choquard type. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1497-1515. doi: 10.3934/dcds.2019064 [16] Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018 [17] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469 [18] T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105 [19] Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574 [20] Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389

2020 Impact Factor: 1.392