Abstract. Let y(h) (t, x) be one solution towith a non-homogeneous term h, and y| (0,T )×∂Ω = 0, where Ω ⊂ R n is a bounded domain. We discuss an inverse problem of determining n(n + 1)/2 unknown functions aij.., h 0 suitably, where Γ0 is an arbitrary subboundary, ∂ν denotes the normal derivative, 0 < θ < T and 0 ∈ N. In the case of 0 = (n + 1) 2 n/2, we prove the Lipschitz stability in the inverse problem if we choose (h1, ..., h 0 ) from a set H ⊂ {C ∞ 0 ((0, T ) × ω)} 0 with an arbitrarily fixed subdomain ω ⊂ Ω. Moreover we can take 0 = (n + 3)n/2 by making special choices for h , 1 ≤ ≤ 0. The proof is based on a Carleman estimate.Mathematics Subject Classification. 35R30, 35K20.
The authors prove Carleman estimates for the Schrödinger equation in Sobolev spaces of negative orders, and use these estimates to prove the uniqueness in the inverse problem of determining L p -potentials. An L 2 -level observability inequality and unique continuation results for the Schrödinger equation are also obtained.
Consider the scattering of the two-or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source scattering problem which is to reconstruct the source function. Our results show that increasing stability can be obtained for the inverse problem by using only the Dirichlet boundary data with multi-frequencies.
In this paper, we derive a Carleman estimate for a stochastic wave equation. Then we apply the Carleman estimate to solve an inverse source problem of determining two kinds of sources simultaneously for a stochastic wave equation. Our main result is a uniqueness result for the inverse source problem.
ABSTRACT. Consider the one-dimensional stochastic Helmholtz equation where the source is assumed to be driven by the white noise. This paper concerns the stability analysis of the inverse random source problem which is to reconstruct the statistical properties of the source such as the mean and variance. Our results show that increasing stability can be obtained for the inverse problem by using suitable boundary data with multifrequencies.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.