An inverse boundary value problem for the p -Laplacian: a linearization approach

Abstract

This work tackles an inverse boundary value problem for a p-Laplace type partial differential equation parametrized by a smoothening parameter τ ≥ 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 < ∞ and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a Hölder continuous conductivity coefficient to the solution of the Neumann problem, is Fréchet differentiable, excluding the degenerate case τ = 0 that corresponds to the classical (weighted) p-Laplace equation. 2010 Mathematics Subject Classification. 65N21, 35J60. Key words and phrases. p-Laplacian, inverse boundary value problem, linearization, Bayesian inversion.

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

“…Our main strategy is to use the linearization technique of Isakov and others [28,29,30,31,32] in dealing with nonlinear equations to decompose the inverse problem to the semilinear radiative transport equation ( 1) into an inverse coefficient problem for the linear transport equation where we reconstruct σ a and σ s by the result of Bal-Jollivet-Jungon [6], and an inverse source problem for the linear transport equation where we reconstruct the two-photon absorption coefficient σ b . This is the same type of strategy that have been successfully employed to solve many inverse problems for nonlinear PDEs recently; see for instance [4,11,13,14,15,18,20,25,26,33,35,36,38,39,40,41,43,44,45,46,47,50,58,61,62,63] and reference therein.…”

Variable exponent Calderón's problem in one dimension

“…By the above, we only need countably many measurements of the DN map, but since the functions s n depend implicitly on the unknown conductivity γ, we have no way to determine which Dirichlet data m to use beforehand. It is not possible to explicitly reconstruct P f using proposition 19.…”

“…This is the first investigation of an inverse problem related to the variable exponent p(•)-Laplacian. The problem in the constant exponent case was introduced by Salo and Zhong [31] in 2012, after which other theoretical results have been published [3,5,7,8,18,24], as well as a numerical study [19]. The works of Sun and others consider the problem of recovering the dependence of A in div (A(x, u, ∇u)∇u) = 0 on all of x, u, and ∇u, but they either do not admit degenerate equations [29,34] or assume the equivalent of constant p [21].…”

“…This is the first investigation of an inverse problem related to the variable exponent p(·)-Laplacian. The problem in the constant exponent case was introduced by Salo and Zhong [31] in 2012, after which other theoretical results have been published [3,5,7,8,18,24], as well as a numerical study [19]. The works of Sun and others consider the problem of recovering the dependence of A in div (A(x, u, ∇u)∇u) = 0 on all of x, u, and ∇u, but they either do not admit degenerate equations [29,34] or assume the equivalent of constant p [21].One can think of the variable exponent conductivity equation − div γ |∇u| p(x)−2 ∇u = 0 as arising from a non-linear Ohm's lawwhich has a power law dependence between the current j and the gradient of the electric potential u at every point, but where the exponent in the power law varies from point to point.…”

Recovering a variable exponent

“…The physical motivation for these is that Ohm's law is only an approximation and many real-world systems exhibit highly non-linear IV (or current-voltage) patterns. Power law -type patterns lead to the p-conductivity equation − div γ |∇u| p−2 ∇u = 0 first introduced by Salo and Zhong [45] and investigated further by Salo and others others [5,6,11,7,9,26,27,32]. In these works the exponent 1 < p < ∞ is assumed to be a known constant, whilst the linear factor in the conductivity γ is the unknown.…”