Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
- Druet, Pierre-Étienne
2010 Mathematics Subject Classification
- 49K20 80A22 53A10 35J25
- Stefan-Gibbs-Thompson problem, Singularity of mean-curvature type, Optimal control, Pointwise gradient state constraints, First order optimality conditions
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.
- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2443--2463.