### A lemma and a conjecture on the cost of rearrangements

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We prove a logarithmic stability estimate for a parabolic inverse problem concerning the localization of unknown cavities in a thermic conducting medium $\Omega $ in ${\mathbb{R}}^{n}$, $n\ge 2$, from a single pair of boundary measurements of temperature and thermal flux.

We prove a logarithmic stability estimate for a parabolic inverse problem concerning the localization of unknown cavities in a thermic conducting medium Ω in ${\mathbb{R}}^{n}$, n ≥ 2, from a single pair of boundary measurements of temperature and thermal flux.

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces...

The boundary trace problem for positive solutions of $$-\mathrm{\u25b3}u+g\left(x,u\right)\ge 0$$ is considered for nonlinearities of absorption type, and three different methods for defining the trace are compared. The boundary trace is obtained as a generalized Borel measure. The associated Dirichlet problem with boundary data in the set of such Borel measures is studied.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T)\times \omega $, where $T\>0$ is a sufficiently large time interval and a subdomain $\omega $ satisfies a non-trapping condition.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.