Compact embeddings of p-Sobolev-like cones of nuclear operators

Abstract:

Let p≥ 2 , Ω ⊆ R^N smooth bounded domain, V∈ L ^∞(Ω) non-negative, and S<inf>1</inf> the space of self-adjoint trace-class operators on L ²(Ω). We prove that W1,p, the p-Sobolev-like cone of operators T∈ S<inf>1</inf> having eigenvalues ν<inf>i</inf>, i∈ N, and an eigenbasis B={ψi/i∈N} of L ²(Ω) such that ∑ <inf>i</inf><inf>∈</inf><inf>N</inf>| ν<inf>i</inf>| ∫ <inf>Ω</inf>[| ∇ ψ<inf>i</inf>| ^p+ V(x) | ψ<inf>i</inf>| ^p] dx< + ∞, is compactly embedded in S<inf>1</inf>. In the path, we prove regularity properties for the density function associated to T as well as Gagliardo–Nirenberg type inequalities departing from Lieb–Thirring type conditions. We apply the compactness property to minimize free energy functionals where the entropy term is generated by a Cassimir-class function related to the eigenvalue problem of the Schrödinger operator - αΔ + V, α> 0 , with Dirichlet condition. Our results extend those previously obtained for p= 2 by Dolbeault et al.

Año de publicación:

2022

Keywords:

• Regularity properties
• Gagliardo–Nirenberg type inequality
• Free-energy functional
• Trace-class operator
• Sobolev-like cones
• Nuclear operator
• Compact embedding

Fuente:

scopus