Recovery of harmonic functions from partial boundary data respecting internal pointwise values

Abstract

We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary

On the data completion problem and the inverse obstacle problem with partial Cauchy data for Laplace’s equation

“…Burman in [19] proposes a regularization through discretization : in particular, he obtains an optimal convergence result of the discretized solution to the exact one. In the particular case of a 2d problem, Leblond et al in [41] use a complex-analytic approach to recover the solution of the Cauchy problem respecting furthermore additional pointwise constraints in the domain. Let us also mention the work of Kozlov et al [39] which presents the classical KMF algorithm used widely for numerical simulations.…”

Экстремальные Задачи С Ограничениями В $H^2$ И Формулы Карлемана

“…reaches a minimum at λ = 0. By the boundedness of b, we may differentiate this function with respect to λ under the integral sign, and equating the derivative to 0 at λ = 0 yields (28).…”

On Some Extremal Problems for Analytic Functions with Constraints on Real or Imaginary Parts

“…Such constrained best approximation problems arise in the context of system theory, for harmonic identification purposes and recovery of transfer functions from partial boundary data, see [13], in Hardy classes of the unit disk (for discrete time systems) or of the right-half plane (for continuous time). They also happen to furnish regularized resolution schemes for overdetermined boundary value problems concerning Laplace or elliptic partial differential equations in domains of dimension 2, see [16,23,27,30]. In these frameworks, data are provided by approximate pointwise values (measurements, corrupted by errors) of a function belonging to some Hardy class or of its real or imaginary part, partially available on a subset of the domain or of its boundary.…”