On Cauchy's problem: I. A variational Steklov–Poincaré theory

Abstract

In 1923 (Lectures on Cauchy's Problem in Linear PDEs (New York, 1953)), J Hadamard considered a particular example to illustrate the ill-posedness of the Cauchy problem for elliptic partial differential equations, which consists in recovering data on the whole boundary of the domain from partial but over-determined measures. He achieved explicit computations for the Laplace operator, due to the squared shape of the domain, to observe, in fine, that the solution does not depend continuously on the given boundary data. The primary subject of this contribution is to extend the result to general domains by proving that the Cauchy problem has a variational formulation that can be put under a (variational) pseudo-differential equation, set on the boundary where the data are missing, and defined by a compact Steklov-Poincaré-type operator. The construction of this operator is based on the Dirichlet-to-Neumann mapping, and its compactness is derived from the elliptic regularity theory. Next, using mathematical tools from the linear operator theory and the convex optimization, we provide a comprehensive analysis of the reduced problem which enables us to state that (i) the set of compatible data, for which existence and uniqueness are guaranteed, is dense in the admissible data space; (ii) when the existence fails, due to possible noisy data, the variational problem can be consistently approximated by the least-squares method, that is the incompatibility measure (the deviation indicator or the variational crime made on the Steklov-Poincaré equation) equals zero though all the minimizing sequences blow up.

Parameter choice in the Lavrentiev regularization of the data completion problem

“…In some places of the subsequent we use the notation Γ CD for Γ C ∪ Γ D and Γ ID for Γ I ∪ Γ D . We resume the simplifying assumptions of [4], Γ C and Γ I are supposed to have no common vertices (in two dimensions) nor common edges (in three dimensions) and are therefore separated by Γ D . Assume be given a flux ϕ C and a datum g C .…”

“…The data completion problem for the Laplace operator is expressed as a Cauchy problem: find u such that The lack of a source term in (1) and the choice of the homogeneous Dirichlet condition (2) are only for clarity and are by no way restrictive. The variational framework at the basis of our analysis has been introduced and analyzed in [4]. A brief description of it follows [3] and needs the introduction of some notations.…”

“…As in [4], when existence occurs, the solution of problem (1-4) is the one whose trace λ on Γ I satisfies u = u D (λ, g C ) = u N (λ, ϕ C ) and solves the Steklov-Poincaré problem:…”

“…Recently (in 2005), a variational framework, based on the Steklov-Poincaré approach, has been led down and extensively analyzed in [4,3]. The results established in there, enable a substantial advances in the knowledge of the qualitative properties of the Cauchy problem.…”

“…Moreover, the convergence rate of the approximated incompatibility measure toward zero turns to be almost quadratic, instead of the expected linear rate. Recall that the minimum value of Kohn-Vogelius function equals zero and is the incompatibility measure of the variational SteklovPoincaré Cauchy problem (see [4]). Using the discrepancy principle for the incompatibility measure, provided by the minimum value of the Kohn-Vogelius function rather than the leastsquares function, is the topic of Section 5.…”