Abstract
In this paper, we are interested in the inverse problem of the determination of the unknown part ∂ Ω , Γ 0 of the boundary of a uniformly Lipschitzian domain Ω included in ℝ N from the measurement of the normal derivative ∂ n v on suitable part Γ 0 of its boundary, where v is the solution of the wave equation ∂ t t v x , t − Δ v x , t + p x v x = 0 in Ω × 0 , T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂ Ω . From necessary conditions, we estimate a Lagrange multiplier k Ω which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.
Highlights
Introduction and MainResult e inverse problem in this paper means the problem of reconstructing object from observation data
By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. e Lipschitz stability is established by increasing of the energy of the system
We prove the existence result of the solution for our inverse problem by determining the optimal shape in Section 2 and we prove the existence of a Lagrange multiplier, which appears in the optimality condition of the problem
Summary
Let (Ωn)n∈N be a sequence of open sets having the ε− cone property and converging to Ω in the sense of Hausdorff. We study the existence of the result of the following optimization problem: inf{J(w), w ∈ O}, where the class of admissible domains is defined by (12). E fact that sequence (Ωn)n∈N ∈ O is bounded ensures the existence of a subsequence (Ωnk)nk∈N ∈ O and a domain Ω ∈ O such that (Ωnk)nk converges to Ω in the sense of Hausdorff according to Lemma 1. Let E: O ⟶ X, with O being the set of domains having the ε− cone property and X being a normal vector space. We recall that this space is identified with the subspace of L∞(RN), whose partial derivatives in the sense of distributions are functions of L∞(RN). Note that |∇v| (− k(Ω))1/2 is an optimality condition and if we situate Γ, we will be able to estimate |∇v| on Γ. erefore, we deduce an approximation of Lagrange multiplier k(Ω)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Mathematics and Mathematical Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
