2017/02/07 by Pierluigi Colli, Colli, Pierluigi, Jürgen Sprekels +1
Computer Science · Materials Science · #35K61 #49J20 #49J50 #74M15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Metallurgy and Material Science #Optimization and Control (math.OC) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1702.01907
openalex publication_date 2017/02/07 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this paper, we study an optimal boundary control problem for a model for\nphase separation taking place in a spatial domain that was introduced by P.\nPodio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a\nstrongly coupled system of nonlinear parabolic differential inclusions, in\nwhich products between the unknown functions and their time derivatives occur\nthat are difficult to handle analytically; the system is complemented by\ninitial and boundary conditions. For the order parameter of the phase\nseparation process, a dynamic boundary condition involving the Laplace-Beltrami\noperator is assumed, which models an additional nonconserving phase transition\noccurring on the surface of the domain. We complement in this paper results\nthat were established in the recent contribution appeared in Evol. Equ. Control\nTheory 6 (2017), pp. 35-58, by the two authors and Gianni Gilardi. In contrast\nto that paper, in which differentiable potentials of logarithmic type were\nconsidered, we investigate here the (more difficult) case of nondifferentiable\npotentials of double obstacle type. For such nonlinearities, the standard\ntechniques of optimal control theory to establish the existence of Lagrange\nmultipliers for the state constraints are known to fail. To overcome these\ndifficulties, we employ the following line of approach: we use the results\ncontained in the preprint arXiv:1609.07046 [math.AP] for the case of\n(differentiable) logarithmic potentials and perform a so-called "deep quench\nlimit". Using compactness and monotonicity arguments, it is shown that this\nstrategy leads to the desired first-order necessary optimality conditions for\nthe case of (nondifferentiable) double obstacle potentials.\n