Calculus of Variations and Geometric Measure Theory

G. Buttazzo - F. Maestre - B. Velichkov

Optimal potentials for problems with changing sign data

created by buttazzo on 19 Oct 2017
modified by velichkov on 04 Nov 2019


Published Paper

Inserted: 19 oct 2017
Last Updated: 4 nov 2019

Journal: Journal of Optimization Theory and Applications
Year: 2017


We consider optimal control problems where the state equation is an elliptic PDE of a Schr\"odinger type, governed by the Laplace operator $-\Delta$ with the addition of a potential $V$, and the control is the potential $V$ itself, that may vary in a suitable admissible class. In a previous paper (Ref. \cite{bgrv14}) an existence result was established under a monotonicity assumption on the cost functional, which occurs if the data do not change sign. In the present paper this sign assumption is removed and the existence of an optimal potential is still valid. Several numerical simulations, made by {\tt FreeFem++}, are shown.

Keywords: shape optimization, capacitary measures, Optimal potentials, Schr\"odinger operators, free boundary