AI 中文总结
针对耦合热弹性板系统的反问题，通过分析输入-输出算子性质，将其转化为Tikhonov泛函极小化问题，建立了极小元的存在性。
AI 中文摘要
我们研究由四阶位移方程和热演化方程组成的耦合热弹性板系统，二者通过空间变化的耦合因子α(x)关联，该模型通过算子div(α(x)∇θ)和div(α(x)∇u_t)描述热弹性相互作用。在u满足齐次Neumann条件、θ满足Dirichlet条件下，我们建立了正问题的适定性，推导了最优能量估计并证明了解对给定数据的连续依赖性。我们进一步引入对应所考虑反问题的输入-输出算子，证明其是紧算子且Lipschitz连续，确认了相关反问题的不适定性。利用这些性质，将反问题表述为Tikhonov泛函的极小化问题，并证明了极小元的存在性。
英文摘要
We investigate a coupled thermoelastic plate system consisting of a fourth-order displacement equation and a heat evolution equation linked through a spatially varying coupling factor $α(x)$. The model accounts for thermoelastic interactions through the operators $\operatorname{div}(α(x)\nabla θ)$ and $\operatorname{div}(α(x)\nabla u_t)$. We establish the well-posedness of the direct problem under homogeneous Neumann conditions for $u$ and Dirichlet conditions for $θ$, deriving optimal energy estimates and demonstrating continuous dependence of solutions on the given data. We further introduce an input-output operator corresponding to the considered inverse problem and show that it is compact and Lipschitz continuous, confirming the ill-posed nature of the associated inverse problem. Using these properties, the inverse problem is formulated as a minimization problem for the Tikhonov functional, and we establish the existence of a minimizer.
