On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms. (10th October 2017)

• Record Type: Journal Article
• Authors: Egger, Herbert; Pietschmann, Jan-Frederik; Schlottbom, Matthias
• Is Part Of: Inverse problems. Volume 33:Number 11(2017:Nov.)
• Publication Date: 2017-10-10
• Subjects: uniqueness -- parameter identification -- nonlinear PDE -- identifiability -- nonlinear diffusion; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa8cae
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider the identification of nonlinear diffusion coefficients of the form a ( t, u ) or a ( u ) in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail.