WIAS Preprint No. 1708, (2012)

Some mathematical problems related to the 2nd order optimal shape of a crystallization interface



Authors

  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500

2010 Mathematics Subject Classification

  • 49K20 80A22 53A10 35J25

Keywords

  • Stefan-Gibbs-Thompson problem, Singularity of mean-curvature type, Optimal control, Pointwise gradient state constraints, First order optimality conditions

Abstract

We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.

Appeared in

  • Discrete Contin. Dyn. Syst., 35 (2015) pp. 2443--2463.

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