## Some nonconvex shape optimization problems.(English)Zbl 0982.49024

Cellina, Arrigo (ed.) et al., Optimal shape design. Lectures given at the joint CIM/CIME summer school, Tróia, Portugal, June 1-6, 1998. Berlin: Springer. Lect. Notes Math. 1740, 7-46 (2000).
The author presents three nonconvex shape optimization problems (and several generalizations of these problems):
a) minimizing paths for the opaque square for which he recalls Theorem 1: For each maximum number $$k\in\mathbb{N}$$ of components and for every convex compact $$D\subset\mathbb{R}^n$$, there exists a curve $$C_k\subset D$$ minimizing one-dimensional Hausdorff-measure under the geometric side constraint, that every straight line intersecting $$D$$ also intersects the curve;
b) Newton’s problem of minimal resistance for which it can be shown Theorem 2: Let $$\Omega\subset\mathbb{R}^n$$ be convex, then there exists a minimizer $$u$$ of $$R$$ in $$C_M\equiv \{v\in W^{1,\infty}_{\text{loc}}(\Omega): 0\leq v\leq M, v\text{ concave}\}$$, where $$R(v)= \int_\Omega {1\over 1+|\nabla v|^2} dx$$; the minimizer is in general not unique, because, if $$\Omega$$ is the unit disk of $$\mathbb{R}^2$$, $$u$$ is not radial; the Euler-Lagrange equation associated with $$R$$ is formally $$-\text{div}({\nabla u\over(1+|\nabla u|^2)^2})= 0$$, it is of mixed (elliptic-hyperbolic) type and holds in $$C^0_M$$;
c) extremal eigenvalue problems for which he remembers Theorem 3: Let $$\lambda_1(\Omega)$$ be the first eigenvalues for $$\Delta u+\lambda u= 0$$ in $$\Omega\subset \mathbb{R}^N$$, $$u= 0$$ on $$\partial\Omega$$, then the problem of minimizing $$\lambda_1$$ among all open sets of prescribed $$N$$-dimensional Lebesgue measure has a ball for solution; let $$\gamma_1(\Omega)$$ be the first eigenvalue for $$\Delta\Delta u-\gamma u=0$$ in $$\Omega\subset \mathbb{R}^2$$, $$u= \partial_nu= 0$$ on $$\partial\Omega$$, then the problem of minimizing $$\gamma_1$$ among all open sets of prescribed 2-dimensional Lebesgue measure has a disc for solution.
For the entire collection see [Zbl 0954.00031].

### MSC:

 49Q10 Optimization of shapes other than minimal surfaces 35J20 Variational methods for second-order elliptic equations 49J20 Existence theories for optimal control problems involving partial differential equations