Elliptic equations: Single boundary measurements

Abstract

In this chapter we consider the elliptic second-order differential equation Au=finΩ,f=f0−∑j=1n∂jfjAu=finΩ,f=f0−∑j=1n∂jfj with the Dirichlet boundary data u=g0on∂Ω.u=g0on∂Ω. We assume that A = div(−a∇) + b ⋅ ∇ + c with bounded and measurable coefficients a (symmetric real-valued (n × n) matrix) and complex-valued b and c in L∞(Ω). Another assumption is that A is an elliptic operator; i.e., there is ɛ0 > 0 such that a(x)ξ ⋅ ξ ≥ ɛ0 | ξ | 2 for any vector ξ∈Rnξ∈Rn and any x ∈ Ω. Unless specified otherwise, we assume that Ω is a bounded domain in RnRn with the boundary of class C2. However, most of the results are valid for Lipschitz boundaries.