Criteria for guaranteed breakdown in two-phase inhomogeneous bodies
  Lower bounds are obtained on the maximum field strength in one or both phases in a body containing two-phases. These bounds only incorporate boundary data that can be obtained from measurements at the surface of the body, and thus may be useful for determining if breakdown has necessarily occurred in one of the phases, or that some other nonlinearities have occurred. It is assumed the response of the phases is linear up to the point of electric, dielectric, or elastic breakdown, or up to the point of the onset of nonlinearities. These bounds are calculated for conductivity, with one or two sets of boundary conditions, for complex conductivity (as appropriate at fixed frequency when the wavelength is much larger than the body, i.e., for quasistatics), and for two-dimensional elasticity. Sometimes the bounds are optimal when the field is constant in one of the phases, and using the algorithm of Kang, Kim, and Milton (2012) a wide variety of inclusion shapes having this property, for appropriately chosen bodies and appropriate boundary conditions, are numerically constructed. Such inclusions are known as $E_\Omega$-inclusions. 
Patrick Bardsley, Michael S. Primrose, Michael Zhao, Jonathan Boyle,   Nathan Briggs, Zoe Koch, and Graeme W. Milton
math.AP