Boundary-Value Problems for the Biharmonic Equation with a Linear Parameter

Abstract

We consider two boundary-value problems for the equation Δ2 u(x, y) - λΔu(x, y) = ƒ(x, y) with a linear parameter on a domain consisting of an infinite strip. These problems are not elliptic boundary-value problems with a parameter and therefore they are non-standard. We show that they are uniquely solvable in the corresponding Sobolev spaces and prove that their generalized resolvent decreases as 1/|λ| at infinity in L2 (ℝ x (0,1)) and W1 2 (ℝ x (0,1)).