Modification of the method of boundary functions for singulary perturbed partial differential equations (Q1337840)

In constructing asymptotic expansions (a.e.) of solutions of boundary problems for singularly perturbed (s.p.) equations, i.e., equations containing small parameters at high derivatives, the method of boundary functions is widely used. In constructing the asymptotics of a solution of a boundary problem in a domain \(\Omega\) for an s.p. partial differential equation by the method of boundary functions as a rule one needs the continuity of the solution in the closed domain \(\overline\Omega\). If the boundary of the domain \(\partial\Omega\) contains corners, then for the solution to be continuous in \(\overline\Omega\) it is necessary that compatibility conditions hold for the boundary data at these points. When the compatibility conditions do not hold at the corners one can only construct the first terms of the asymptotics by the method of boundary functions. In the present paper we propose a modification of the method of boundary functions, with the help of which, in a number of cases one can construct uniform a.e. of solutions of s.p. boundary problems up to any order without compatibility conditions at corners of the boundary.