Some mathematical problems related to the second order optimal shape of a crystallisation interface
Pierre‐Étienne Druet · Discrete and Continuous Dynamical Systems · 2015
We consider the problem to optimise 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 minimisation principle of the free energy while the temperature is solving the heat equation with radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallisation, we can expect that the interface has a global representation as a graph. 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 derivatives of the surface and for the surface temperature gradient.