Stable reconstruction of regular 1-Harmonic maps with a given trace at the boundary

Abstract

We consider the numerical solvability of the Dirichlet problem for the 1-Laplacian in a planar domain endowed with a metric conformal with the Euclidean one. Provided that a regular solution exists, we present a globally convergent method to find it. The global convergence allows to show a local stability in the Dirichlet problem for the 1-Laplacian nearby regular solutions. Such problems occur in conductivity imaging, when knowledge of the magnitude of the current density field (generated by an imposed boundary voltage) is available inside. Numerical experiments illustrate the feasibility of the convergent algorithm in the context of the conductivity imaging problem.

Keywords:

• 1-Laplacian
• current density impedance imaging
• characteristics
• global convergence
• boundary value problems

AMS Subject Classifications:

• 35R30
• 35J60
• 31A25

Additional information

Funding

The work of the first and third authors was supported by the NSF [grant number DMS-1312883].