General non local boundary value problem for second order elliptic equation

Abstract

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper [1]. Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well-known Dore-Venni theorem. K E Y W O R D SDore-Venni theorem, non local boundary conditions, operator-differential equation, UMD spaces M S C ( 2 0 1 0 ) INTRODUCTIONRobin boundary conditions are very well investigated. It is not the case for non local Robin boundary conditions, in general, non local boundary conditions have got less attention.The operator in operator differential equation can be differential-difference operator as in [19,20]. The operators  and  in boundary conditions may correpond to multidimentional non local boundary conditions. Such problems arise in several concrete physical phenomena. For example, they arise in plasma theory, in the theory of multidimentional diffusion process [19,21], in heat transfer subject to mass specification, in electro-chemistry ([4-6]). They appear, also, in reduced fluid-structure interaction appearing in hemodynamics application ([17]).We are interested here in this kind of problems. The novelty in this paper is that we consider the two boundary conditions as non local Robin conditions. Similar problems with local Robin conditions have been considered in ([7,8]).An extensive study of non local boundary value problems can be found in ([13,19,22]).In this paper, is a complex Banach space, belongs to (0, 1; ) where 1 < < ∞, 0 and 1 are given elements of ; ,  and  are closed linear operators in . Let us consider the second-order operational differential equation in ′′ ( ) +  ( ) = ( ), a.e. ∈ (0, 1), (1.1) with non-local boundary conditions { ′ (1) −  (0) = 0 , ′ (0) +  (1) = 1 , (1.2)where , , , ∈ ℂ.( , ) ≠ (0, 0), ( , ) ≠ (0, 0), ( , ) ≠ (0, 0), ( , ) ≠ (0, 0).(1.3) 1470