Existence of weakly neutral coated inclusions of general shape in two dimensions

ABSTRACT

A two-dimensional inclusion of core–shell structure is neutral to multiple uniform fields if and only if the core and the shell are concentric disks, provided that the conductivity of the matrix is isotropic. An inclusion is said to be neutral if upon its insertion the uniform field is not perturbed at all. In this paper, we consider inclusions of core–shell structure of general shape which are weakly neutral to multiple uniform fields. An inclusion is said to be weakly neutral if the field perturbation is mild. We show, by an implicit function theorem, that if the core is a small perturbation of a disk, then we can coat it by a shell so that the resulting structure becomes weakly neutral to multiple uniform fields.

Keywords:

• Neutral inclusion
• weakly neutral inclusion
• core–shell structure
• polarization tensor vanishing structure
• implicit function theorem

2010 Mathematics Subject Classifications:

• 35J47
• 45P05

Acknowledgments

This work was supported by the NRF grants number 2016R1A2B4011304 and 2017R1A4A1014735, JSPS KAKENHI grant number JP16K13768 and by A3 Foresight Program among China (NSF), Japan (JSPS), and Korea (NRF 2014K2A2A6000567).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Research Foundation of Korea [grants numbers 2016R1A2B4011304 and 2017R1A4A1014735], by A3 Foresight Program among China (NSF), Japan (JSPS), and Korea (NRF) (2014K2A2A6000567), and by the National Natural Science Foundation of China (NSF of China) [grant number 11901523].