An inverse boundary value problem for the \(p\)-Laplacian

This work tackles an inverse boundary value problem for a \(p\)-Laplace type partial differential equation parametrized by a smoothening parameter \(\tau \geq 0\). The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. The numerical studies are based on a straightforward linearization of the forward map, and they demonstrate that the accuracy of such an approach depends nontrivially on \(1 < p < \infty\) and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a H\"older continuous conductivity coefficient to the solution of the Neumann problem, is Fréchet differentiable, excluding the degenerate case \(\tau=0\) that corresponds to the classical (weighted) \(p\)-Laplace equation.