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Regularity and singularities of optimal convex shapes in the plane. (English) Zbl 1268.49051

Let \(S_{ad}\) be a set of convex admissible subsets of \({\mathbb R}^2\), \(J:S_{ad}\to {\mathbb R}\) a shape functional, and \(M:S_{ad}\to{\mathbb R}^d\) an extra constraint. This article is concerned with the regularity of a convex minimizer \(\Omega_0\subset {\mathbb R}^2\) of \(J\) on \(S_{ad}\), possibly with the additional condition \(M(\Omega)=M_0\in {\mathbb R}^d\), under appropriate assumptions on \(S_{ad}\) and \(J\). The authors use an analytic approach representing every convex set in polar coordinates as \(\Omega=\{(r,\theta): ru(\theta)<1\}\), where \(u\in W^{1,\infty}({\mathbb T})\), \({\mathbb T}=[0,2\pi)\), is the gauge function of \(\Omega\) and \(W^{1,\infty}({\mathbb T})\) is the set of \(2\pi\)-periodic \(W^{1,\infty}_{loc}({\mathbb R})\)-functions. The convexity of \(\Omega\) is equivalent to the weak inequality \(u''+u\geq 0\). A new functional \(j(u)=J(\Omega_u)\), a subset \(F_{ad}=\{u\in W^{1,\infty}(T):\Omega_u\in S_{ad}\}\), and a new constraint \(m(u)=M(\Omega_u)\) are then defined.
So the problem is reduced to the one of the regularity of a function \(u_0\in W^{1,\infty}({\mathbb T})\) satisfying either \[ j(u_0)=\min\{j(u): u''+u\geq 0, u\in F_{ad}\}, \tag\# \] or \[ j(u_0)=\min\{j(u): u''+u\geq 0, u\in F_{ad}, m(u)=M_0\}, \tag\#\# \] where \(j:W^{1,\infty}({\mathbb T})\to {\mathbb R}\), \(m:W^{1,\infty}({\mathbb T})\to {\mathbb R}^d\), \(M_0\in {\mathbb R}^d\). In both cases, the set \(F_{ad}\) is chosen as \[ F_{ad}=\{u\in W^{1,\infty}({\mathbb T}): k_1\leq u\leq k_2, u>0\}, \tag{*} \] where \(k_1, k_2:{\mathbb T}\to \overline{{\mathbb R}_+}\) are upper and lower semicontinuous functions, respectively.
The main results in the paper are of two types: in Theorems 1 and 2, it is proved that the optimal shapes are regular using the first optimality condition for the minimization problem. Similar arguments may be found in [G. Carlier, J. Nonlinear Convex Anal. 3, No. 2, 125–143 (2002; Zbl 1030.49023)]. In both Theorems the functional \(j(u)\) is assumed to be of the form \[ j(u)=r(u)+\int_{{\mathbb T}} G(\theta,u(\theta),u'(\theta))\,d\theta, \] where \(r\) and \(G\) satisfy
(i) \(r:W^{1,\infty}({\mathbb T})\to {\mathbb R}\) is \(C^1\) around \(u_0\) and \(G:(\theta,u,q)\in{\mathbb T}\times (0,\infty)\times {\mathbb R}\to {\mathbb R}\) is \(C^2\) around \({\mathbb T}\times u_0({\mathbb T})\times\roman{Conv}(u_0'({\mathbb T}))\),
(ii) \(r'(u_0)\in L^p({\mathbb T})\) for some \(p\in [1,\infty]\),
(iii) \(G_{qq}>0\) in \({\mathbb T}\times u_0({\mathbb T})\times\roman{Conv}(u_0'({\mathbb T}))\).
In Theorem 1 it is shown that if \(u_0>0\) satisfies (#) with \(F_{ad}\) given by (*), then \(u_0\in W^{2,p}({\mathbb T}_{in})\), where the set \({\mathbb T}_{in}\) is defined by \({\mathbb T}_{in}=\{\theta\in {\mathbb T}: k_1(\theta)<u(\theta)<k_2(\theta)\}\).
In Theorem 2, if \(u_0>0\) satisfies (##) under the assumptions of Theorem 1 and the additional condition that \(m:W^{1,\infty}({\mathbb T})\to {\mathbb R}\) is \(C^1\) around \(u_0\) with \(m'(u_0)\in (L^p({\mathbb T}))^d\), then \(u_0\in W^{2,p}({\mathbb T}_{in})\). Proofs of Theorems 1 and 2 are given in Section 3.1, and explicit examples are shown in Section 3.2.
In the second type of results, Theorems 3 and 4, it is proved that the optimal shapes are polygons using second order optimality conditions, (see [J. Lamboley and A. Novruzy, SIAM J. Control Optim. 48, No. 5, 3003–3025 (2009; Zbl 1202.49053)] and [T. Lachand-Robert and M. A. Peletier, Math. Nachr. 226, 153–176 (2001; Zbl 1048.49011)]). More precisely, in Theorem 3 it is shown that given a solution \(u_0>0\) of (#) with \(F_{ad}\) as in (*), and \(j:W^{1,\infty}({\mathbb T})\to {\mathbb R}\) a functional which is \(C^2\) around \(u_0\) satisfying that \(\exists s\in [0,1)\), \(\alpha>0\), \((\beta,\gamma)\in [0,\infty)^2\), such that \(\forall v\in W^{1,\infty}({\mathbb T})\) \[ j''(u_0)(v,v)\leq -\alpha |v|^2_{H^1({\mathbb T})}+\gamma |v|_{H^1({\mathbb T})} ||v||_{H^s({\mathbb T})}+\beta ||v||_{H^s({\mathbb T})}^2, \] then \(u_0''+u_0\) is a finite sum of Dirac masses in a connected component \(I\) of \({\mathbb T}_{in}\), (and hence the boundary of the associated convex set is locally a polygon).
In Theorem 4, assuming that \(u_0>0\) satisfies (##) under the assumptions of Theorem 3, and the additional hypotheses \(j'(u_0)\in (C^0({\mathbb T}))'\), \(m:W^{1,\infty}\to {\mathbb R}^d\) is \(C^2\) around \(u_0\), \(m'(u_0)\in (C^0({\mathbb T})')^d\) is onto, and \(||m''(u_0)(v,v)||\leq\beta' ||v||^2_{H^s({\mathbb T})}\), for some \(\beta'\in {\mathbb R}\), \(s\in [0,1)\), we have that \(u_0''+u_0\) is a finite sum of Dirac masses in any connected component of \({\mathbb T}_{in}\). Proofs of Theorems 3 and 4 are given in Section 4.1 and examples in Section 4.2.
Some remarks and perspectives can be found in Section 5.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
52A40 Inequalities and extremum problems involving convexity in convex geometry
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References:

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