2016/07/07 by Pierluigi Colli, Colli, Pierluigi, Gianni Gilardi +3
Materials Science · #35K55 #49K20 #74A15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1607.01991
openalex publication_date 2016/07/07 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
This paper is concerned with a distributed optimal control problem for a\nnonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version\nof a model for two-species phase segregation on an atomic lattice under the\npresence of diffusion. The local model has been investigated in a series of\npapers by P. Podio-Guidugli and the present authors; the nonlocal model studied\nhere consists of a highly nonlinear parabolic equation coupled to an ordinary\ndifferential inclusion of subdifferential type. The inclusion originates from a\nfree energy containing the indicator function of the interval in which the\norder parameter of the phase segregation attains its values. It also contains a\nnonlocal term modeling long-range interactions. Due to the strong nonlinear\ncouplings between the state variables (which even involve products with time\nderivatives), the analysis of the state system is difficult. In addition, the\npresence of the differential inclusion is the reason that standard arguments of\noptimal control theory cannot be applied to guarantee the existence of Lagrange\nmultipliers. In this paper, we employ recent results proved for smooth\nlogarithmic potentials and perform a so-called `deep quench' approximation to\nestablish existence and first-order necessary optimality conditions for the\nnonsmooth case of the double obstacle potential.\n