发表机构
The Hong Kong Polytechnic University; Ningbo University（香港理工大学; 宁波大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对热方程逆源问题，在低Hausdorff维数集合上的边界通量或内部观测下，建立了条件稳定性，提出了新的边界谱不等式，并获得了对数或双对数稳定性结果。
AI 中文摘要
本文研究有界$C^{1,1}$区域内热方程逆源问题的条件稳定性，其中时间因子已知而空间分量未知。我们关注支撑在低Hausdorff维数集合上的观测，并在此设定下建立条件稳定性。对于具有正$q$维Hausdorff容度的紧集上的边界观测，我们从全时边界通量观测建立对数稳定性，并从延迟时间边界通量观测建立双对数稳定性。当观测集包含在平坦边界片内时，允许的维数范围为$q>d-2$；在一般$C^{1,1}$边界上为$q>d-1-c_{d+1}$，其中$c_{d+1}>0$仅依赖于维数。推导这些结果的一个关键要素是Dirichlet拉普拉斯算子的新边界谱不等式，该不等式通过此类边界集上其椭圆延拓的法向导数的观测来控制有限Dirichlet谱和。我们的结果还涵盖在具有正$q$维Hausdorff容度（对于某个$q>d-1$）的集合上进行内部观测的逆热源问题，对于$H_0^1(\Omega)$中的一般源，从全时观测获得对数稳定性；对于适当谱Gevrey类中的源，从终端时间观测获得Hölder稳定性。
英文摘要
This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(Ω)$ and Hölder stability from terminal-time observations for sources in a suitable spectral Gevrey class.
Comments27 pages
