Optimization problems with fixed volume constraints and stability results related to rearrangement classes

Liu, Yichen; Emamizadeh, Behrouz; Farjudian, Amin

Abstract

The material in this paper has been divided into two main parts. In the first part we describe two optimization problems—one maximization and one minimization—related to a sharp trace inequality that was recently ob- tained by G. Auchmuty. In both problems the admissible set is the one comprising characteristic functions whose supports have a fixed measure. We prove the maximization to be solvable, whilst the minimization will turn out not to be solvable in general. We will also discuss the case of radial do- mains. In the second part of the paper, we study approximation and stability results regarding rearrangement optimization problems. First, we show that if a sequence of the generators of rearrangement classes converges, then the corresponding sequence of the optimal solutions will also converge. Second, a stability result regarding the Hausdorff distance between the weak closures of two rearrangement classes is presented.

Citation

Liu, Y., Emamizadeh, B., & Farjudian, A. (2016). Optimization problems with fixed volume constraints and stability results related to rearrangement classes. Journal of Mathematical Analysis and Applications, 443(2), https://doi.org/10.1016/j.jmaa.2016.06.017

• Journal Article Type: Article
• Acceptance Date: Jun 1, 2016
• Publication Date: Nov 15, 2016
• Deposit Date: Apr 6, 2018
• Publicly Available Date: Nov 16, 2018
• Journal: Journal of Mathematical Analysis and Applications
• Print ISSN: 0022-247X
• Electronic ISSN: 0022-247X
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 443
• Issue: 2
• DOI: https://doi.org/10.1016/j.jmaa.2016.06.017
• Keywords: Trace inequality, Boundary value problem, Maximization, Mini- mization, Approximation, Stability, Rearrangement theory
• Public URL: https://nottingham-repository.worktribe.com/output/828522
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0022247X16302475?via%3Dihub