文摘
For the solution to \(\partial ^2_tu(x,t)-\triangle u(x,t)+q(x)u(x,t)=\delta (x,t)\) and \(u\mid _{t<0}=0\), consider an inverse problem of determining \(q(x), x\in \Omega \) from data \(f=u\mid _{S_T}\) and \(g=(\partial u/\partial \mathbf {n})\mid _{S_T}\). Here \(\Omega \subset \{(x_1,x_2,x_3)\in \mathbb {R}^3\mid x_1>0\}\) is a bounded domain, \(S_{T}=\{(x,t)\mid x\in {\partial \Omega },\vert {x}\vert<t<T+\vert {x}\vert \}\), \(\mathbf {n}=\mathbf {n}(x)\) is the outward unit normal \(\mathbf {n}\) to \(\partial \Omega \), and \(T>0\). For suitable \(T>0\), prove a Lipschitz stability estimation: $$\begin{aligned} \left\| {q_1-q_2}\right\| _{L^2(\Omega )}\le C\left\{ \left\| {f_1-f_2}\right\| _{H^1(S_T)}+\left\| {g_1-g_2} \right\| _{L^2(S_T)}\right\} , \end{aligned}$$provided that \(q_1\) satisfies a priori uniform boundedness conditions and \(q_2\) satisfies a priori uniform smallness conditions, where \(u_k\) is the solution to problem (1.1) with \(q = q_k, k = 1, 2\).KeywordsInverse problemStabilityCarleman estimateHyperbolic equationMathematics Subject Classification35R3035R2535L10References1.Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)MATHGoogle Scholar2.Baudouin, L., De Buhan, M., Ervedoza, S.: Global Carleman estimates for waves and applications. Commun. Partial Differ. Equ. 38, 823–859 (2013)MathSciNetCrossRefMATHGoogle Scholar3.Beilina, L., Klibanov, M.V.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. 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Pures Appl. 78, 65–98 (1999)MathSciNetCrossRefMATHGoogle ScholarCopyright information© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsXue Qin1Email authorShumin Li11.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina About this article CrossMark Print ISSN 2194-6701 Online ISSN 2194-671X Publisher Name Springer Berlin Heidelberg About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s40304-016-0091-4_A Stability Estimate for an Invers", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s40304-016-0091-4_A Stability Estimate for an Invers", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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