Sobolev-like cones of trace-class operators on unbounded domains: Interpolation inequalities and compactness properties

Abstract

In this paper we extend the compactness properties for trace-class operators obtained by Dolbeault, Felmer and Mayorga-Zambrano to a smooth unbounded domain Ω⊆Rd, d≥3. We consider V, a non-negative potential on Ω that blows up at infinity, and the normed space HV(Ω)={u∈H01(Ω):‖u‖V2=∫Ω(∣∇u(x)∣2+∣u(x)∣2V(x))dx<∞}. A positive self-adjoint trace-class operator R belongs to the Sobolev-like cone HV,+1 if (ψi,R)N⊆HV(Ω) and 《R》V=∑i=1∞νi,R‖ψi,R‖V2<∞, where (νi,R)i∈N is the sequence of occupation numbers of R and (ψi,R)i∈N⊆L2(Ω) is a corresponding Hilbertian basis of eigenfunctions. We prove that a sequence in HV,+1, bounded in energy 《⋅》V, has a subsequence that converges in trace norm; this is analogous to the classical Sobolev immersion H1(Ω)⊆L2(Ω). We prove the existence of lower bounds for nonlinear free energy functionals and, by doing so, we establish Lieb–Thirring type inequalities as well as some Gagliardo–Nirenberg type interpolation inequalities; then our compactness result is applied to minimize nonlinear free energy functionals working on HV,+1.