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 ℋ V,+1 if (ψi,R)ℕ⊆H V(Ω) and 〈 〈〈R〉〉V= Σi=1∞νi,R∥ψ i,R∥V2< ∞, where (νi,R)i ∈ ℕ is the sequence of occupation numbers of R and (ψi, R)∈ ℕ⊆L 2(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 H 1(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. © 2013 Elsevier Ltd. All rights reserved.

Año de publicación:

2013

Keywords:

• Schrödinger operator
• Trace-class operator
• compactness
• Free energy

Fuente:

scopus