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Shape optimization problems over classes of convex domains. (English) Zbl 0921.49030

The authors study shape optimization problems of the general form \[ \min_{A\in {\mathcal A}}\int_{\partial A}f(x,\nu(x))d{\mathcal H}^{n-1}, \] where \({\mathcal A}\) is a class of admissible convex domains, \(\nu(x)\) denotes the outer normal to \(\partial A\), and \(f\) is continuous. An example is provided by Newton’s model of resistance of a body with a flat bottom. If the body occupies \(\{(x,y)\mid 0\leq y\leq u(x)\), \(x\in\Omega \subset\mathbb{R}^2\}\), then \(f(x, \nu (x))=(\nu_3(x)^+)^3\). If \({\mathcal A}\) satisfies certain side constraints, e.g., if members of \({\mathcal A}\) have volume bounded away from zero and are contained in some compact set \(Q\subset\mathbb{R}^n\), then the minimization problem has a solution. An essential ingredient in the proof is a theorem of Yu. G. Reshetnyak [Sib. Mat. Zh. 9, 1039-1045 (1968; Zbl 0176.44402); ibid. 1386-1394 (1968; Zbl 0169.18301)].
Reviewer: B.Kawohl (Köln)

MSC:

49Q10 Optimization of shapes other than minimal surfaces
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