Abstract
Geometric evolution equations, such as mean curvature flow and surface diffusion, play an important role in mathematical modeling in various fields, ranging from materials to life science. Controlling the surface or interface evolution would be desirable for many of these applications. We attack this problem by considering the bulk contribution, which defines a driving force for the geometric evolution equation, as a distributed control. In order to solve the control problem we use a phase-field approximation and demonstrate the applicability of the approach on various examples. In the first example the effect of an electric field on the evolution of nanostructures on crystalline surfaces is considered. The mathematical problem corresponds to surface diffusion or a Cahn-Hilliard model. In the second example we consider mean curvature flow or a Allen-Cahn model
Mathematics Subject Classification (2000). 35R35; 49M.
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Haußer, F., Janssen, S., Voigt, A. (2012). Control of Nanostructures through Electric Fields and Related Free Boundary Problems. In: Leugering, G., et al. Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics, vol 160. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0133-1_29
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DOI: https://doi.org/10.1007/978-3-0348-0133-1_29
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Publisher Name: Birkhäuser, Basel
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Online ISBN: 978-3-0348-0133-1
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