Phase Transitions with the Line Tension Effect: the Super-quadratic Case

Received (Day Month Year) Communicated by (xxxxxxxxxx) Let Ω be an open bounded set of R 3 and let W and V be two non-negative continuous functions vanishing at α, β and α ′ , β ′ , respectively. We analyze the asymptotic behavior as ε → 0, in terms of Γ-convergence, of the following functional Fε(u) := ε p−2 Z Ω |Du| p dx + 1 ε p−2 p−1 Z Ω W (u)dx + 1 ε Z ∂Ω V (T u)dH 2 (p > 2), where u is a scalar density function and T u denotes its trace on ∂Ω. We show that the singular limit of the energies Fε leads to a coupled problem of bulk and surface phase transitions.