Augmenting a Fredholm Operator of Zero Index to Achieve Invertibility for Elliptic Boundary Value Problems

Abstract

An elliptic partial differential operator satisfies the Gårding inequality, which leads to a Fredholm operator when the boundary conditions are also properly posed. In many applications, this Fredholm operator has zero index. Therefore n orthogonality (or, compatibility) conditions must be satisfied by the data, and the solutions have n degrees of freedom-they are nonunique. In order to fix those n degrees of freedom for the uniqueness of the solution, n accessory linear conditions are usually prescribed. This leads to an augmented system. In this paper, we formulate a simple operator-theoretic theorem to enable us to characterize when the augmented linear system is uniquely solvable. We apply this theorem to the Neumann problem as well as to the traction boundary value problem in elastostatics based upon a simple layer potential boundary integral approach. Criteria of unique solvability are established for several types of accessory conditions for such boundary value problems.