{"title":"Shape optimization problems in control form","authors":"G. Buttazzo, Francesco Paolo Maiale, B. Velichkov","doi":"10.4171/rlm/942","DOIUrl":null,"url":null,"abstract":"We consider a shape optimization problem written in the optimal control form: the governing operator is the p-Laplacian in the Euclidean space R, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case p > d, where the existence occurs under very mild conditions, and the case p ≤ d, where additional assumptions have to be made on the data.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Rendiconti Lincei-Matematica e Applicazioni","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/rlm/942","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
We consider a shape optimization problem written in the optimal control form: the governing operator is the p-Laplacian in the Euclidean space R, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case p > d, where the existence occurs under very mild conditions, and the case p ≤ d, where additional assumptions have to be made on the data.
期刊介绍:
The journal is dedicated to the publication of high-quality peer-reviewed surveys, research papers and preliminary announcements of important results from all fields of mathematics and its applications.