Abstract.
We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of
where
The sets \(D\subset \Omega\), \(\Gamma\subset \overline{\Omega}\), 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 \(\vert\nabla u\vert^2\), then \(\partial \Omega\) is \(C^\infty\) 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.
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Received: 5 March 2002, Accepted: 3 September 2002, Published online: 17 December 2002
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Chambolle, A., Larsen, C.J. \(C^\infty\) regularity of the free boundary for a two-dimensional optimal compliance problem. Cal Var 18, 77–94 (2003). https://doi.org/10.1007/s00526-002-0181-x
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DOI: https://doi.org/10.1007/s00526-002-0181-x