The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential
Michael Bildhauer, Martin Fuchs, V. G. Osmolovskiî · Mathematical Methods in the Applied Sciences · 2001
Abstract We consider the problem of minimizing among functionsu:ℝd⊃Ω→ℝd,u∣∂Ω=0, and measurable subsetsEof Ω. Herefh+,f−denote quadratic potentials defined on Ω¯×{symmetricd×dmatrices},his the minimum energy offh+andε(u) is the symmetric gradient of the displacement fieldu. An equilibrium stateû,ÊofJ(u,E) is called one‐phase ifE=∅︁ orE=Ω, two‐phase otherwise. For two‐phase states,σ∣∂E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parametershϵℝ,σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of infJ(u,E) in the limitσ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.