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Regularity of minimizers of shape optimization problems involving perimeter. (English. French summary) Zbl 1381.49044

If \({\mathcal A}\) is a class of domains in \(\mathbb R^d\) and \(J:{\mathcal A}\to\mathbb R\) is a given shape functional, then a shape optimization problems is to minimize \(\{J(\Omega);\;\Omega\in{\mathcal A}\}\). The most important example of a shape optimization is the question of minimizing the perimeter \(J=P\) defined as \(P(D)={\mathcal H}^{d-1}(\partial\Omega)\) when \(D\) is smooth under volume constraint. The well-known isoperimetric inequality asserts that the ball is the unique minimizer for this problem. In more general situations, for example for the constrained isoperimetric problem \(\min\{P(\Omega);\;|\Omega|=m\;\Omega\subset D\}\), where \(D\) is a box in \(\mathbb R^d\) too narrow to contain a ball of volume \(m\), the regularity issue is not trivial. It was proved that, if \(D\) is bounded, an optimal shape \(\Omega^*\) exists in the class of sets of finite perimeter and that \(\partial\Omega\cap D\) is smooth if \(d\leq 7\). An optimal shape \(\Omega^*\) is called a quasi-minimizer of the perimeter if there exist \(C\in\mathbb R\), \(a\in(d-1,d]\), and \(r_0>0\) such that for every ball \(B_r\) with \(r\leq r_0\), \(P(\Omega^*)\leq P(\Omega)+C r^\alpha\) such that \(\Omega\Delta\Omega^*\subset B_r\cap D\) for any \(\Omega\). In [ESAIM Control Optim. Calc. Var. 10, 99–122 (2004; Zbl 1118.35078)], T. Briançon studied the regularity of optimal shapes of the Dirichlet energy \(E_f\) with volume constraint, where \(E_f(\Omega)=\min\limits_u\left\{\int\limits_\Omega\left(\frac12|\nabla u|^2-f u\right)dx\right\}\), \(\Omega\subset D\), \(\Omega\) is open, and \(|\Omega|=a>0\). The author proved that the optimal shapes have \(C^{1,\alpha}\) regularity if \(f\geq 0\). In [Appl. Math. Optim. 69, No. 2, 199–231 (2014; Zbl 1297.49077)], G. De Philippis and B. Velichkov considered the existence and regularity of a solution of the shape optimization problem \( \min \left\{\lambda_k(\Omega)\right\}\) such that \(\Omega\subset\mathbb{R}^d\), \(\Omega\) is open, \(P(\Omega)=1\), and \(|\Omega|<\infty\), where \(\lambda_k\) denotes the \(k\)-th eigenvalue of the Dirichlet Laplacian, that is, the \(k\)-th smallest positive real number such that the equation \(-\Delta u_k = \lambda_k(\Omega) u_k\) has a nontrivial solution, where \(u_k\in H^1_0(\Omega)\). They showed that every solution \(\Omega\) is a bounded connected open set whose boundary is \(C^{1,\alpha}\) outside a closed set of Hausdorff dimension \(d-8\). Analogous results were proved also for more general spectral functionals of the form \(F(\Omega)=f(\lambda_{k_1}(\Omega), \ldots , \lambda_{k_p}(\Omega))\) for increasing functions \(f\:\mathbb{R}^p\to\mathbb{R}\) satisfying a bi-Lipschitz type condition.
In this paper, the authors prove the existence and regularity of optimal shapes for the problem \(\min\{P(\Omega)+\mathcal{G}(\Omega);\;\Omega\subset D\;\wedge\;|\Omega|=m\}\), where \(P\) denotes the perimeter, \(|\cdot|\) is the volume, and the functional \(\mathcal{G}\) is either the Dirichlet energy \(E_f\) with respect to a function \(f\in L^p\), or a spectral functional of the form \(F(\lambda_1,\dots,\lambda_k)\), where \(\lambda_k\) is the \(k\)th eigenvalue of the Dirichlet Laplacian and \(F:\mathbb R^k\to\mathbb R\) is locally Lipschitz continuous and increasing in each variable. They show that the solution of the problem is the domain \(D\) that is either the whole space \(\mathbb R^d\) or a bounded domain. Also, they show that every solution \(\Omega^*\) of the problem is bounded and it is a quasi-minimizer of the perimeter with exponent \(d-d/p\) or \(d\), respectively.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49N60 Regularity of solutions in optimal control
35P99 Spectral theory and eigenvalue problems for partial differential equations
35R35 Free boundary problems for PDEs
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