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Well-posedness, control and computation of a one-phase Stefan problem with Neumann condition*

Published online by Cambridge University Press:  14 November 2011

Goong Chen
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.
Shunhua Sun
Department of Mathematics, Sichuan University, Chengdu, Sichuan, People's Republic of China
Quan Zheng
Department of Mathematics, Shanghai University of Science and Technology, Shanghai, People's Republic of China


A one-phase Stefan problem can be reduced to an equivalent variational inequality by using the Baiocchi-Duvaut transformation. In this paper, we study the variational inequality by formulating it as a set-valued partial differential equation. The existence of solutions is proved by applying a generalized Schauder fixed fixed point theorem for set-valued mappings. Uniqueness and regularity of solutions are also obtained. In §3, we regard the boundary value Neumann data as boundary controls and combine both the variational inequality and the classical approaches to study the effects of controls on the free boundary and the state (i.e. temperature). In §4, we further use the theory to study an optimal “ice-melting” problem. Our results show that if the controls have fixed total input heat flux and are constrained in magnitude, then the optimal control is “bang-bang”. If the admissible controls are not constrained in magnitude, then the optimal control is a Dirac delta type distribution which is no longer admissible. In the last section, our existence theory is combined with the finite difference method and non-linear programming techniques to obtain numerical solutions.

Research Article
Copyright © Royal Society of Edinburgh 1984

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