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Erschienen in:
Inverse Problems Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

4. Elliptic Equations: Single Boundary Measurements

verfasst von : Victor Isakov

Erschienen in: Inverse Problems for Partial Differential Equations

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel Uniqueness and Stability in the Cauchy Problem
Nächstes Kapitel Elliptic Equations: Many Boundary Measurements
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Metadaten
Titel
Elliptic Equations: Single Boundary Measurements
verfasst von
Victor Isakov
Copyright-Jahr
2017
Buch
Inverse Problems for Partial Differential Equations

Print ISBN: 978-3-319-51657-8

Electronic ISBN: 978-3-319-51658-5

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-51658-5_4