Small-parameter asymptotic expansions of the solution of a degenerate problem (Q1178128)

The boundary value problem in the rectangle \(Q=\{ 0\leq x\leq 1\), \(| y|\leq 1\}\) for an elliptic equation with small parameter \(\varepsilon\) is considered: \[ \varepsilon \Delta u- a(x,y)\partial_ y u+ k(x,y)u= f(x,y), \quad x\in Q; \qquad u(x,\varepsilon)=0, \quad x\in \partial Q. \] It is supposed that the coefficient \(a\) satisfies \(a(x,\phi(x))=0\) for some \(\phi\in C^ \infty\). There are two different cases. The first one corresponds to the simplest example when \(a(x,y)\equiv y\). The full asymptotic expansion of the solution \(u(x,y,\varepsilon)\) as \(\varepsilon\to 0\) is constructed for this problem. The second case corresponds to the example \(a(x,y) \equiv-y\). For this problem only the first \(N\) terms of the asymptotic expansion of the solution \(u(x,y,\varepsilon)\) as \(\varepsilon\to 0\) are constructed. The number \(N\) depends on the relation \(k(x,y)/ a(x,y)\).