Concentration phenomena for the volume functional in unbounded domains: identification of concentration points

We study the variational problemSεF(Ω)=1ε2∗ sup ∫ΩF(u) : ∫Ω|∇u|2⩽ε2, u=0 on ∂Ωin possibly unbounded domains Ω⊂Rn, where n⩾3, 2∗=2nn−2 and F satisfies 0⩽F(t)⩽α|t|2∗ and is upper semicontinuous. Extending earlier results for bounded domains, we show that (almost) maximizers of SεF(Ω) concentrate at a harmonic center, i.e. a minimum point of the Robin function τΩ (the regular part of the Green function restricted to the diagonal). Moreover, we obtain the asymptotic expansionSεF(Ω)=SF1−nn−2 w∞2 minΩ̄ τΩε2+o(ε2),where SF and w∞ depend only on F but not on Ω and can be computed from radial maximizers of the corresponding problem in Rn. The crucial point is to find a suitable definition of τΩ(∞). Interestingly the correct definition may be different from the lower semicontinuous extension of τΩ|Ω̄⧹{∞} to ∞, at least for n⩾5