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We consider Cheeger-like shape optimization problems of the form $$minbig{|Omega|^alpha J(Omega) : Omegasubset Dbig}$$ where $D$ is a given bounded domain and $alpha$ is above the natural scaling. We show the existence of a solution and analyze as $J(Omega)$ the particular cases of the compliance functional $C(Omega)$ and of the first eigenvalue $lambda_1(Omega)$ of the Dirichlet Laplacian. We prove that optimal sets are open and we obtain some necessary conditions of optimality.
We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Con
We suggest a new greedy strategy for convex optimization in Banach spaces and prove its convergent rates under a suitable behavior of the modulus of uniform smoothness of the objective function.
In this paper, optimal actuator shape for nonlinear parabolic systems is discussed. The system under study is an abstract differential equation with a locally Lipschitz nonlinear part. A quadratic cost on the state and input of the system is consider
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on
In this paper, we propose some new proximal quasi-Newton methods with line search or without line search for a special class of nonsmooth multiobjective optimization problems, where each objective function is the sum of a twice continuously different