Boundary determination for hybrid imaging from a single measurement

We recover the conductivity σ at the boundary of a domain from a combination of Dirichlet and Neumann boundary data and generalized power/current density data at the boundary, from a single quite arbitrary set of data, in AET or CDII. The argument is elementary, algebraic and local. More generally, we consider the variable exponent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {p(\,·\,)} -Laplacian as a forward model with the interior density data <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msup> </m:mrow> </m:math> {σ|∇ u|q} , and find out that single measurement specifies the boundary conductivity when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {p-q≥ 1} , and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both p and q may depend on the spatial variable x , but they are assumed to be a priori known. We illustrate the practical situations with numerical examples with the code available.