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## Optimal Control of a Semilinear PDE with Nonlocal Radiation Interface Conditions

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Author(s) : Christian Meyer , Peter Philip , Fredi Tröltzsch

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 33-2004

MSC 2000

49K20 Problems involving partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
49J20 Optimal control problems involving partial differential equations
80M50 Optimization

Abstract :
We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The problem arises from the aim to optimize the temperature gradient within crystal growth by the physical vapor transport (PVT) method. Based on a minimum principle for the semilinear equation as well as $L^\infty$-estimates for the weak solution, we establish the existence of an optimal solution as well as necessary optimality conditions. The theoretical results are illustrated by results of numerical computations.

Keywords : Optimal control, semilinear elliptic equations, nonlocal interface conditions, boundedness of solutions