Nonlocal boundary value problem for second order abstract elliptic differential equation

Mohamed Denche

doi:10.1155/s1085337599000135

Abstract

We establish conditions that guarantee Fredholm solvability in the Banach space Lp of nonlocal boundary value problems for elliptic abstract differential equations of the second order in an interval. Moreover, in the space L2 we prove in addition the coercive solvability, and the completeness of root functions (eigenfunctions and associated functions). The obtained results are then applied to the study of a nonlocal boundary value problem for Laplace equation in a cylindrical domain.

Highlights

Fredholm property of boundary value problems is investigated in [1, 2, 3] for elliptic partial differential equations, and in [4, 13, 14, 15] for abstract differential equations.In this paper, we establish conditions guaranteeing that nonlocal boundary value problems for elliptic partial differential equations of the second order in an interval are Fredholm solvable in the Banach spaces Lp
We establish conditions guaranteeing that nonlocal boundary value problems for elliptic partial differential equations of the second order in an interval are Fredholm solvable in the Banach spaces Lp
For the solution of the considered problem we prove the noncoercive estimates

Summary

Introduction

Fredholm property of boundary value problems is investigated in [1, 2, 3] for elliptic partial differential equations, and in [4, 13, 14, 15] for abstract differential equations. We establish conditions guaranteeing that nonlocal boundary value problems for elliptic partial differential equations of the second order in an interval are Fredholm solvable in the Banach spaces Lp. For the solution of the considered problem we prove the noncoercive estimates. The completeness of root functions of regular boundary value problems was proved in [1, 5, 7, 10, 13]. 2. Necessary notations and definitions Let H be a Hilbert space, A a linear closed operator in H and D(A) its domain. 154 Second order abstract elliptic differential equation norm, and by Lp((0, 1), H ) the Banach space of strongly measurable functions x → u(x) : (0, 1) → H, whose pth power is summable, with the norm u p 0,p.

Solvability of the principal problem

Fredholm solvability of general problem

Application