Shape derivatives for minima of integral functionals

• 成果类型: Article
• 署名作者: Bouchitte, Guy; Fragala, Ilaria; Lucardesi, Ilaria
• 署名单位: Universite de Toulon; Polytechnic University of Milan
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-013-0712-6
• 发表日期: 2014
• 页码: 111-142
• 关键词: bounded slope condition; optimization; CONVERGENCE; calculus; validity; domain

摘要

For varying among open bounded sets in , we consider shape functionals defined as the infimum over a Sobolev space of an integral energy of the kind , under Dirichlet or Neumann conditions on . Under fairly weak assumptions on the integrands and , we prove that, when a given domain is deformed into a one-parameter family of domains through an initial velocity field , the corresponding shape derivative of at in the direction of exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of on . Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.