The stability for the Cauchy problem for elliptic equations

G Alessandrini, L Rondi, E Rosset, S Vessella - Inverse problems, 2009‏ - iopscience.iop.org
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 …

Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates

S Vessella - Inverse Problems, 2008‏ - iopscience.iop.org
In this article, we review the main results concerning the issue of stability for the
determination of unknown boundary portions of a thermic conducting body from Cauchy …

Observability inequalities and measurable sets.

J Apraiz, L Escauriaza, G Wang, C Zhang - Journal of the European …, 2014‏ - ems.press
This paper presents two observability inequalities for the heat equation over×(0, T). In the
first one, the observation is from a subset of positive measure in×(0, T), while in the second …

[PDF][PDF] Optimal stability for inverse elliptic boundary value problems with unknown boundaries

G Alessandrini, E Beretta, E Rosset… - Annali della Scuola …, 2000‏ - numdam.org
In this paper we study a class of inverse problems associated to elliptic boundary value
problems. More precisely, those inverse problems in which the role of the unknown is played …

The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

A Logunov, E Malinnikova, N Nadirashvili… - … and Functional Analysis, 2021‏ - Springer
Let Ω Ω be a bounded domain in R^ n R n with C^ 1 C 1 boundary and let u_ λ u λ be a
Dirichlet Laplace eigenfunction in Ω Ω with eigenvalue λ λ. We show that the (n-1)(n-1) …

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

X Tolsa - Communications on Pure and Applied Mathematics, 2023‏ - Wiley Online Library
Let be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In
this paper it is shown that if u is a function harmonic in and continuous in, which vanishes in …

Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations

A De Luca, V Felli, S Vita - Advances in Mathematics, 2022‏ - Elsevier
We study local asymptotics of solutions to fractional elliptic equations at boundary points,
under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a …

Unique continuation principles for a higher order fractional Laplace equation

V Felli, A Ferrero - Nonlinearity, 2020‏ - iopscience.iop.org
In this paper we prove the strong unique continuation principle and the unique continuation
from sets of positive measure for solutions of a higher order fractional Laplace equation in …

Boundary Unique Continuation on -Dini Domains and the Size of the Singular Set

C Kenig, Z Zhao - Archive for Rational Mechanics and Analysis, 2022‏ - Springer
Let u be a harmonic function in a C 1-Dini domain D⊂ R d such that u vanishes on a
boundary surface ball∂ D∩ B 5 R (0). We consider an effective version of its singular set …

Examples of non-Dini domains with large singular sets

C Kenig, Z Zhao - Advanced Nonlinear Studies, 2023‏ - degruyter.com
Let u be a nontrivial harmonic function in a domain D⊂ R d, which vanishes on an open set
of the boundary. In a recent article, we showed that if D is a C 1-Dini domain, then, within the …