An Inverse Problem for a Semilinear Elliptic Equation on Conformally Transversally Anisotropic Manifolds

IF 2.4 1区数学 Q1 MATHEMATICS Annals of Pde Pub Date : 2023-06-27 DOI:10.1007/s40818-023-00153-w

Ali Feizmohammadi, Tony Liimatainen, Yi-Hsuan Lin

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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.

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共形横各向异性流形上的一个半线性椭圆型方程的反问题

给定一个共形横向各向异性流形（M，g），我们考虑了半线性椭圆方程$$\beart｛aligned｝（-\Delta_｛g｝+V）u+qu^2＝0\quad\hbox｛on｝\M\end｛align｝$$我们证明了先验未知光滑函数q可以根据与该方程相关的Dirichlet到Neumann映射的知识唯一确定。这扩展了Feizmohammadi和Oksanen（J Differ Equ 269（6）：4683–47192020），Lassas等人（J Math Pures Appl 145:44–821021）的先前已知结果。我们的证明是基于对方程的过微分：我们将方程线性化到比非线性的二阶更高的阶，并为线性化引入非消失边界迹。我们研究线性化方程的所谓高斯拟模解的两个或多个乘积的相互作用。我们发展了一种渐近演算来求解拉普拉斯方程，这些方程将这些相互作用作为源项。

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