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

Citations:

Zbl 0176.44402; Zbl 0169.18301
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