Uniqueness of a solution to the stationary problem in kinetics of the growth of thin films from the gas phase on solid supports (Q2714016)

The system NEWLINE\[NEWLINE \frac{\partial}{\partial x_i}\left\{a_{ij}(1 - Q^2) \frac{\partial}{\partial x_j}\left[\frac{Q}{b(1-Q)}e^{-aQ}\right]\right\} + \lambda(1-Q)\left[d - \frac{Q}{b(1-Q)}e^{-aQ}\right] = 0, NEWLINE\]NEWLINE has received considerable attention in the literature [see \textit{A.~V.~Bogdanov, G.~V.~Dubrovskij, M.~P.~Krutikov}, Interactions of gases with surfaces. Detailed description of elementary processes and kinetics, Heidelberg (1995)] as a prototype for the growth of thin films from gas phase on solid supports. Here \(0\leq Q(x)\leq 1\), \(x\in\mathbb R^m\), is the probability of replacing the cell with coordinate \(x\) by an atom. The model constants \(a\), \(b\), \(d\), and \(\lambda\) are assumed positive.NEWLINENEWLINE The main objective of the article is to expose sufficient conditions which guarantee the uniqueness of solutions to the system in the class of periodic functions in \(x\) and to prove a unique solvability theorem for the Dirichlet problem for a small \(\lambda\).