Optimal measures for nonlinear cost functionals (Q853845)

The author studies the following optimization problem \[ \inf_{\mu\in m_0(\Omega)} \Biggl\{J(\mu)= \int_\Omega j(x, u_\mu)\,dx+ \alpha\mu(\Omega)\Biggr\}, \] where \(M_0(\Omega)\) is a subset of a Barel measures which are absolutely continuous, \(\alpha\) is a nonnegative parameter \(\Omega\subset\mathbb{R}^N\) is a bounded domain, \(j: \Omega\times \mathbb{R}\to\mathbb{R}\), \(u_\mu\) is the solution for the state equation \[ \begin{cases} -\Delta u+ u_\mu= f, &\text{in }\Omega\\ u= 0, &\text{on }\partial\Omega\end{cases} \] and \(f\in L^2(\Omega)\). The state equation is a relaxed form of a classical Dirichlet problem and the function to minimize (cost function) is nonlinear. The main results obtained by the author are: a regularity results for the optimal measure and a description of it and the important numerical examples.