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].


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