The stability for the Cauchy problem for elliptic equations

Abstract

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations.We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions.As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.

Smoothness dependent stability in corrosion detection

“…By the local stability estimates for the Cauchy problem discussed in [7,Theorem 1.7] and the bounds established earlier in Theorem 3.1 and Theorem 3.4, we get that for any x0 ∈ Γ r 0 I and any 0 < r < r1 we have…”

Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

“…In this section we formulate a result that we will refer to as the propagation of smallness of the Cauchy data for elliptic PDE. See Lemma 4.3 in [11] and Theorem 1.7 in [1] for the proof of the result below, which we bring not in full generality but in a convenient way for our purposes.…”

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

“…What then remains is a data completion problem for the displacement u in the elliptic equation Au = γ ∇T , with known right-hand side, where from displacement and pseudo-traction ( t = σ(u) n = t + γT| Γ1 n) values on Γ 1 , one has to find the correct boundary function T 0 on Γ 2 (since the pseudo-traction on Γ 1 will depend on T 0 ) to match the given displacement on Γ 2 . This type of data completion is a classical Cauchy problem for an elliptic equation, which is well-known to be ill-posed with respect to the noise in the data, see further [1]. In a more technical language, one can build on this to reformulate problem (2.1), (2.2), (2.7)-(2.9) as an operator equation on the boundary with the linear operator having an unbounded inverse.…”