Regresar

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

Abstract:

Let p≥ 2 , Ω ⊆ RN smooth bounded domain, V∈ L ∞(Ω) non-negative, and S1 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∈ S1 having eigenvalues νi, i∈ N, and an eigenbasis B={ψi/i∈N} of L 2(Ω) such that ∑ i∈N| νi| ∫ Ω[| ∇ ψi| p+ V(x) | ψi| p] dx< + ∞, is compactly embedded in S1. 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.