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
DOI
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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