Identifiability at the boundary for first-order terms

Abstract

Abstract.Let Ω be a domain in R n whose boundary is C 1 if n ≥ 3 or C 1,β if n = 2. We consider a magnetic Schrödinger operator L W,q in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L W,q . We also consider a steady state heat equation with convection term ∆+2W ·∇ and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

“…All these results are nonconstructive. We also mention the recent papers [2], [6] and [28] which consider boundary determination, partial Cauchy data and stability for this inverse problem. In this paper we give a constructive algorithm for recovering curl W and q from Λ W,q .…”

Determining nonsmooth first order terms from partial boundary measurements

“…The related inverse scattering problem has been studied in [7]. We also mention [1] and [22] which consider boundary determination and stability for magnetic Schrödinger operators, and [10] which proves a partial data result for the nonsmooth conductivity equation.…”

An Inverse Problem for the $p$-Laplacian: Boundary Determination

“…We prove Theorem 1.2 by showing that one can perturb these p-harmonic complex exponentials to become solutions of the equation involving γ, concentrating near a boundary point. The proof is similar to the arguments of [11], [12] in the linear case and is actually not that difficult, making use of basic facts such as wellposedness for the Dirichlet problem, inequalities for pth powers of vectors, and Hardy's inequality. We then show Theorem 1.1 by replacing the p-harmonic complex exponentials with certain real valued p-harmonic functions, introduced by Wolff [42], having similar properties as exponentials which allow the proof to go through.…”