The Vishik–Lysternik method in general elliptic boundary value problems with small parameter

This paper gives a survey on the concept of ellipticity with small parameter for general elliptic boundary value problems with operator and boundary conditions depending polynomially on a small parameter. We combine the methods of the theory of general elliptic boundary value problems with the Vishik-Lyusternik method of exponential boundary layer. The main result includes necessary and sufficient conditions for the existence of an a priori estimate of the problem uniform with respect to the parameter. These conditions are formulated in terms of interior and boundary symbols of the problem with parameter introduced in this paper.