Elliptic Equations: Single Boundary Measurements

Abstract

In this chapter we consider the elliptic second-order differential equation $Au = f\quad \mathrm{in}\;\varOmega,f = f_{0} -\sum \limits _{j=1}^{n}\partial _{j}f_{j}$ with the Dirichlet boundary data $u = g_{0}\quad \mathrm{on}\;\partial \varOmega.$ 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 ɛ ₀ > 0 such that a(x)ξ ⋅ ξ ≥ ɛ ₀ | ξ | ² for any vector $\xi \in \mathbb{R}^{n}$ and any x ∈ Ω. Unless specified otherwise, we assume that Ω is a bounded domain in $\mathbb{R}^{n}$ with the boundary of class C ². However, most of the results are valid for Lipschitz boundaries.