{"title":"An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds","authors":"Ali Feizmohammadi, Tony Liimatainen, Yi-Hsuan Lin","doi":"10.1007/s40818-023-00153-w","DOIUrl":null,"url":null,"abstract":"<div><p>Given a conformally transversally anisotropic manifold (<i>M</i>, <i>g</i>), we consider the semilinear elliptic equation </p><div><div><span>$$\\begin{aligned} (-\\Delta _{g}+V)u+qu^2=0\\quad \\hbox { on}\\ M. \\end{aligned}$$</span></div></div><p>We show that an a priori unknown smooth function <i>q</i> can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity <span>\\(qu^2\\)</span>, and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-023-00153-w.pdf","citationCount":"10","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pde","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40818-023-00153-w","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 10
Abstract
Given a conformally transversally anisotropic manifold (M, g), we consider the semilinear elliptic equation
$$\begin{aligned} (-\Delta _{g}+V)u+qu^2=0\quad \hbox { on}\ M. \end{aligned}$$
We show that an a priori unknown smooth function q can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity \(qu^2\), and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.