On an elliptic boundary value problem not in divergence form

Abstract

Let G G be a bounded domain in R n ( n ⩾ 2 ) {R^n}(n \geqslant 2) with smooth boundary ∂ G \partial G . We consider the boundary value problem M u − c u = f Mu - cu = f on G G , u = 0 u = 0 on ∂ G \partial G , where M M is an elliptic differential operator not in divergence form. We discuss the characterization of the first eigenvalue λ 0 {\lambda _0} of M M and the solvability of the boundary value problem in terms of the relationship between c ( ⋅ ) c( \cdot ) and λ 0 {\lambda _0} .