A functional with both bulk and interfacial surface energy is considered. It corresponds to the energy dissipated inside a two-phase electrical conductor in the presence of an electrical contact resistance at the two-phase interface. The effect of embedding a highly conducting particle into a matrix of lesser conductivity is investigated. We find the criterion that determines when the increase in surface energy matches or exceeds the reduction in bulk energy associated with the particle. This criterion is general and applies to any particle with Lipschitz continuous boundary. It is given in terms of the of the second Stekloff eigenvalue of the particle. This result provides the means for selecting energy-minimizing configurations.
Publication Source (Journal or Book title)
SIAM Journal on Mathematical Analysis
Lipton, R. (1998). The second Stekloff eigenvalue and energy dissipation inequalities for functionals with surface energy. SIAM Journal on Mathematical Analysis, 29 (3), 673-680. https://doi.org/10.1137/S0036141096310144