Compact embeddings of \(p\)-Sobolev-like cones of nuclear operators
type: Item
instance of: scholarly article
title: Compact embeddings of \(p\)-Sobolev-like cones of nuclear operators
zbMATH Open document ID: 1501.47033
author: Juan Mayorga-Zambrano
author: Josué Castillo-Jaramillo
author: Juan Burbano-Gallegos
published in: Banach Journal of Mathematical Analysis
publication date: 2022-01-27T00:00:00Z
review text: The \(p\)-Sobolev-like cone, denoted by \(\mathcal{W}^{1,p}\), is the set of nuclear operators that possess eigenvalues \(v_i\), \(i\in \mathbb{N}\), and their corresponding eigenvectors \(\psi_i\) satisfying \(\sum_{i\in \mathbb{N}}|v_i|\int_{\Omega}[|\nabla \psi_i|^p+V(x)|\psi_i|^p]dx <\infty\), where \(\Omega\) is a smooth bounded subset of \(\mathbb{R}^N\) In this paper, it is proved that \(\mathcal{W}^{1,p}\) is compact for \(p\ge2\). This result extends the compactness of \(\mathcal{W}^{1,2}\) obtained by \textit{J. Dolbeault} et al. [Monatsh. Math. 155, No. 1, 43--66 (2008; Zbl 1151.81014)]. Such extension is done following the conventional argument based on the Hölder inequality. Consequently, the compactness of \(\mathcal{W}^{1,p}\) is used to show that a certain type of free energy functionals have operator ground states.
reviewed by: Miyeon Kwon
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47L07
zbMATH DE Number: 7465454
zbMATH Keywords: compact embedding
zbMATH Keywords: nuclear operator
zbMATH Keywords: trace-class operator
zbMATH Keywords: Sobolev-like cones
zbMATH Keywords: Gagliardo-Nirenberg-type inequality
zbMATH Keywords: free-energy functional
zbMATH Keywords: regularity properties
MaRDI profile type: Publication
full work available at URL: s43037-021-00175-1
OpenAlex ID: W4210356438
cites work: Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional \(p\)-Laplacian
cites work: Thermal effects in gravitational Hartree systems
cites work: Functional analysis, Sobolev spaces and partial differential equations
cites work: Compactness properties for trace-class operators and applications to quantum mechanics
cites work: Lieb--Thirring type inequalities and Gagliardo--Nirenberg inequalities for systems
cites work: Q4399897
cites work: Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system
cites work: Sobolev-like cones of trace-class operators on unbounded domains: interpolation inequalities and compactness properties
cites work: Q5653193
cites work: Q4066072
cites work: Q3857325
cites work: Q3913305
cites work: Weak Sufficient Conditions for Fatou's Lemma and Lebesgue's Dominated Convergence Theorem
DOI: 10.1007/S43037-021-00175-1
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description: scientific article; zbMATH DE number 7465454