| Contents |
| The talk is concerned with an optimal control problems for an Allen-Cahn equation with nonlinear dynamic boundary condition involving the Laplace--Beltrami operator. The nonlinearities both in the bulk and on the boundary are assumed to be singular, i.e., they may range from the derivative of logarithmic potentials confined in $ [-1,1]$ to the subdifferential of the indicator function of the interval $ [-1,+1]$ up to a concave perturbation. We address both the cases of distributed and boundary controls.
We first examine the case of logarithmic nonlinearities: in a recent paper by Colli and Sprekels the corresponding control problems were studied, and results concerning existence and first-order necessary and second-order sufficient optimality conditions were shown. Then, in the case of double obstacle potentials, a recent joint work with M. H. Farshbaf-Shaker and J. Sprekels investigated a ``deep quench'' approximation (i.e., approximating the indicator function by logarithmic nonlinearities) and led us to establish both the existence of optimal control and first-order necessary optimality conditions. |
|