Improved bounds for the overall properties of a nonlinear composite dielectric

The construction of generalized Hashin‐Shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material. In recent work, a nonlinear comparison medium has been used; however, this requires detailed knowledge of the properties of the trial fields that are employed. The fields used have the property of ‘bounded mean oscillation’ and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled. Some recent results concerning Riesz transforms and the Beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds. In addition, an exact solution is established for a particular class of composites.