The generalized solution of ill-posed boundary problem

Abstract

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces W 0 k,p (Θ, Σ). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.