The First Boundary Value and Eigenvalue Problems for Degenerate Elliptic Equations, I

which is degenerate on the boundary. In this paper we discuss the first boundary value and eigenvalue problems for the elliptic equations of the same form which may degenerate in the interior of the domain. The equations treated in this paper include as their special type uniformly elliptic equations. We treat only weak solutions. However, we weaken the restriction on the coefficients. Our method to solve the problems owes to Sobolev [2], in which we find the boundary value and eigenvalue problems for the Laplace equation. In § 1 we arrange some inequalities to be used in the succeeding sections. Section 2 is devoted to some basic lemmas applied to a variational method. We solve, by a variational method, the first boundary value and eigenvalue problems in § 3 and § 4, respectively. The author wishes to express his sincere thanks to Prof. M. Hukuhara for his helpful suggestions and constant encouragement.