On The Mixed-Boundary Value Problem In The Equilibrium State Of The Model For The Isothermal Solidification Process Of A Binary Alloy

In this paper we study the mixed-boundary value problem in the equilibrium state of a phase-field model for the isothermal solidification process of a binary alloy. The equations of the problem read:The parameter c accounts for the relative density of the alloy, the parameter φ accounts for the solidification state of the alloy and is equal to 0 in the solid phase and equals 1 in the liquid phase.We investigate the non-degenerate case of the problem, prove the existence of the solution under a L~∞ estimate, and give a maximal principal to the problem. This paper is organized in the following way :In section 1 we make a regularization of the problem by a cut-off method first, and prove the existence of the solution of the regularized problem with the fixed point theorem, then we prove the solution of the regularized problem is also solution of the original problem after an estimate. In section 2, we prove the uniqueness of the solution for the non-degenerate case of the problem. Finally, in section 3 we establish a maximum principle under extra assumptions on the non-linear terms.