带有多个未知参数的非线性基尔霍夫板方程的逆问题
Inverse problems for nonlinear Kirchhoff plate equations with multiple unknown parameters
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中文总结 AI 辅助
本文研究非线性基尔霍夫板方程的逆边值问题，建立正问题整体适定性，针对被动、主动测量情形分别实现初始数据、初始数据与系数的恢复，提出Runge逼近等方法，可推广至多类板模型。
中文摘要 AI 辅助
本文针对（非线性）基尔霍夫板方程在多种一般情形下的逆边值问题提供了全面的研究。我们首先建立了非线性正问题的整体适定性，这不仅为后续的逆分析奠定了基础，也具有独立的理论意义。随后针对被动和主动两种测量情形研究了逆问题：在单次被动边界测量下，我们证明了未知初始数据的稳定恢复；在具有无穷多次边界测量的主动情形下，我们的结果分为两部分：对于具有一般时变势（允许空间无界）的线性方程，我们证明了初始数据和系数的同时恢复；对于非线性方程（其中非线性项和初始数据均未知），我们开发了一种新颖的Runge逼近方法，结合精心构造的几何光学解和围绕非零解的高阶线性化，以证明二者的同时确定。此外，我们引入了一种精细的截断技术，为处理初始数据消失的情形提供了另一种手段。值得注意的是，本文开发的方法和结果可轻易推广到其他边界条件和板模型，包括经典的欧拉-伯努利方程。
英文摘要
This paper provides a comprehensive treatment of inverse boundary value problems for (nonlinear) Kirchhoff plate equations under diverse general settings. We begin by establishing the global well-posedness of the nonlinear forward equations, which not only underpins the subsequent inverse analysis but also holds independent theoretical significance. The inverse problems are then examined for both passive and active measurement regimes. With a single passive boundary measurement, we establish the stable recovery of the unknown initial data. In the active regime with infinitely many boundary measurements, our results are twofold. For linear equations featuring generic time-dependent potentials-allowing for spatial unboundedness, we demonstrate the simultaneous recovery of both initial data and coefficients. For nonlinear equations, where both the nonlinearity and initial data are unknown, we develop a novel Runge approximation approach, together with carefully constructed geometric optics solutions and higher-order linearization around nonzero solutions, to prove their simultaneous determination. Furthermore, we introduce a delicate cut-off technique that provides an alternative means of addressing the scenario of vanishing initial data. Notably, the methodologies and results developed herein are readily generalizable to other boundary conditions and plate models, including the classical Euler-Bernoulli equation.
