The two-variable technique for singular partial differential problems and its justification

Abstract

Dirichlet problems with a small parameter in factor of the highest derivative are considered for bounded domains. A two-variable technique is formalized in order to carry out the study of the main boundary layer. A “secular “ hypothesis is made, and a unique and uniformly valid asymptotic expansion is obtained. However, it is shown that the “secular” hypothesis may be weakened and this yields a whole set of expansions. Then the asymptotic validity of each expansion in the set can be proven for second-order operators by means of an extension of a theorem due to Eckhaus and Jager.