$C^infty$ regularity of the free boundary for a two-dimensional optimal compliance problem期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

$C^infty$ regularity of the free boundary for a two-dimensional optimal compliance problem

Authors:

Antonin?Chambolle,Christopher?J.?Larsen author-information" > author-information__contact u-icon-before" > mailto:cjlarsen@wpi.edu" title=" cjlarsen@wpi.edu" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author

Affiliation:

(1) CEREMADE (CNRS UMR 7534), Université de Paris-Dauphine, 75775 Paris CEDEX 16, France;(2) Department of Mathematical Sciences, Worcester Polytechnic Institute, MA 01609 Worcester, USA

Abstract:

We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of

$mathcal{E}(Omega) = J(Omega) + lambda vertOmegavert + mu mathcal{H}^1(partial Omega)$

where

$J(Omega) = -2 inf left{frac{1}{2} int_{Omega} {bf A} e(u) : e(u)- int_Gamma fcdot u : uin LD(Omega), uequiv 0 textrm{on} D right}.$

The sets

, and the force f are given. We show that if we consider only scalar valued u and constant ${bf A}$

, or if we consider the elastic energy $vertnabla uvert^2$

, then

away from where $Omega$

is pinned. In the scalar case, we also show that, for any ${bf A}$

of class $C^{k,theta}$ , $partial Omega$

is $C^{ k+2,theta}$ . The proofs rely on a notion of weak outward curvature of $partial Omega$

, which we can bound without considering properties of the minimizing fields, together with a bootstrap argument.Received: 5 March 2002, Accepted: 3 September 2002, Published online: 17 December 2002

Keywords:

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