On a free boundary problem modelling thermal oxidation of silicon (Q1190757)

This paper addresses the issue of proving the existence and uniqueness of classical solutions to the one-dimensional (in space) non-equilibrium two-phase Stefan problem. The result is achieved in two steps. First, an auxiliary problem is formulated, in which the domain \(\Omega_ t=(0,b(t))\) is assumed to be given. By obtaining estimates on the solution \(v(x,t)\) it is shown that the auxiliary problem has a unique and bounded solution. The second step entails defining an operator \(F\) which associates with each given \(b\) the corresponding solution to the auxiliary problem, according to \(F(b(t))=\int_ 0^ t v(v(t),t)dt+1\). The operator is shown to be pre-compact and continuous on \(E=\{b\in C^ 1[0,T]\): \(b(0)=1\), \(0\leq \dot b(t)\leq K\}\), and the existence of a solution to the original problem follows with the use of the Schauder fixed point theorem. Then uniqueness is proved separately.