TY - JOUR
T1 - Optimization problems with fixed volume constraints and stability results related to rearrangement classes
AU - Liu, Yichen
AU - Emamizadeh, Behrouz
AU - Farjudian, Amin
N1 - Funding Information:
The first author was supported under a PhD scholarship (Ref no.: PGRS-11-01-01 ) provided by Xi'an Jiaotong-Liverpool University (XJTLU) when part of this paper is completed and he would like to thank XJTLU for its support in this research.
Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2016/11/15
Y1 - 2016/11/15
N2 - 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 obtained 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 domains. 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.
AB - 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 obtained 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 domains. 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.
KW - Approximation
KW - Boundary value problem
KW - Optimization
KW - Rearrangement theory
KW - Stability
KW - Trace inequality
UR - http://www.scopus.com/inward/record.url?scp=84975508936&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2016.06.017
DO - 10.1016/j.jmaa.2016.06.017
M3 - Article
AN - SCOPUS:84975508936
SN - 0022-247X
VL - 443
SP - 1293
EP - 1310
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -