The stability for the Cauchy problem for elliptic equations

By Giovanni Alessandrini, Luca Rondi, Edi Rosset, Sergio Vessella. 2009

360 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6a0fb43ad262ec17acd6f1693595ba99a1a0f643

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Strong unique continuation for general elliptic equations in 2D (2010, result): “In previous studies, [4] by Escauriaza and the author and [8] by Rondi, Rosset, Vessella and the author, it has been ascertained that, when lower order terms are absent, or when the operator L is in the self–adjoint form Lu = −div (A∇u)+du, with A symmetric, the representation Theorem 3.”
Quantitative Runge Approximation and Inverse Problems (2017, background): “2(Γ) kh L2(D 1) δ (16) , where C≥ 1 and δ∈ (0,1) depend on D1,D2,G,Γ,n,K. Proof. We write Ω = D2 \D1. Since L∗w= 0 in Ω, the quantitative unique continuation result of (15) follows for instance from [ARRV09]. Indeed, [ARRV09, Theorem 1.9] implies that whenever E≥ kwkH1(Ω) and η≥ k∂νwkH−1/2(Γ), one h”
Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure (2016, background): “See Lemma 4.3 in [13] and Theorem 1.7 in [2] for the proof of the result below, which we bring not in full generality but in a convenient way for our purposes.”
The fractional Calderón problem: Low regularity and stability (2017, methodology): “ of Proposition 4.1 in [Ru15] and the arguments in [B13] for more details). The interior three balls inequality follows from three balls estimates for uniformly elliptic equations (c.f. for instance [ARRV09]). Proof of Proposition5.4. The result of Proposition5.4follows by rescaling from a three bal”
Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs (2020, background): “Such stability estimates have been widely investigated in the theoretical PDE literature and can be derived for a variety of linear and non-linear PDEs, for instance from the well-known three-balls inequalities [1] and references therein, and Carleman estimates [21] and references therein.”
Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients (2015, background): “, [2,35], one can choose the constant C depending only on Ω, the elliptic and Lipschitz constants of a, the boundedness of a and σ.”