Asymptotics of solutions of some elliptic equations in unbounded domains

A. M. Il'in ; E. F. Lelikova

Sbornik. Mathematics, Tome 47 (1984) no. 2, pp. 295-313

Résumé

This paper considers a boundary value problem for the equation $Lu\equiv((-1)^m P_{2m}(D_x,D_y)+D_y)u=f(x,y)$ in some conical domains $\Omega$, where $x\in\mathbf R^{n-1}$, $y\in\mathbf R^1$, $P_{2m}$ is a homogeneous polynomial of degree $2m$ with real coefficients, and $P_{2m}(\xi,\eta)\geqslant\mu(|\xi|^{2m}+\eta^{2m})$. An essential restriction on the domain is the following condition: the boundary contains no rays parallel to the $y$-axis. The first part of the paper studies, for a wide class of domains $\Omega$, the asymptotics of a fundamental solution and the solution of a boundary value problem subject to the condition that the right-hand side and the boundary data tend rapidly to zero at infinity. In § 3, for a specific domain $\Omega$ and $n=2$, a more involved case is examined, in which the right-hand side and the boundary data are unbounded. Bibliography: 13 titles.