发表机构
Beijing Institute of Technology; City University of Hong Kong（北京理工大学; 香港城市大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文在$W^{2,\infty}$流形上建立Dirichlet与Neumann特征函数的定量边界谱不等式，进而证明热方程解的边界可观测性及由边界法向导数唯一确定初态，并给出最优阶对数稳定性估计。
AI 中文摘要
设 $(\mathcal{M},g)$ 为具有非空边界和一致椭圆 Lipschitz 度量的紧致连通 $W^{2,\infty}$ Riemann 流形。我们针对 Dirichlet 和 Neumann 特征函数，从 $\partial\mathcal{M}\times(-s_0,s_0)$ 的具有正曲面测度的任意可测子集建立了定量边界谱不等式。此外，我们应用 Dirichlet 谱不等式建立了边界可观测性估计 \\[ \\|u(T)\\|_{L^2(\mathcal{M})}^2 \le C \int_J \\|\partial_{\nu_g}u(x,t)\\|^2 \\,\mathrm{d}S_g\\,\mathrm{d}t. \\] 其中 $J\subset\partial\mathcal{M}\times(0,T)$ 是可测的且具有正曲面-时间测度。一个仅需在观测边界补丁附近具有 $W^{2,\infty}$ 正则性的局部定理作为独立推论被保留。可观测性不等式的证明结合了来自正测度边界集的定量延拓与低频谱集中。该结果表明热方程解的边界可观测性也可在 $W^{2,\infty}$ 域中实现。最后，作为应用，我们证明了在 $J$ 上对边界法向导数的测量能唯一确定热方程的初始状态。在 $D(-\Delta_{g, D}^{s/2})$ 中的先验界下，恢复满足最优阶 $s/2$ 的对数稳定性估计。
英文摘要
Let $(\mathcal{M},g)$ be a compact connected $W^{2,\infty}$ Riemannian manifold with nonempty boundary and a uniformly elliptic Lipschitz metric. We establish quantitative boundary spectral inequalities for both Dirichlet and Neumann eigenfunctions from arbitrary measurable subsets of $\partial\mathcal{M}\times(-s_0,s_0)$ of positive surface measure. Moreover, we apply the Dirichlet spectral inequality to establish the boundary observability estimate \[ \|u(T)\|_{L^2(\mathcal{M})}^2 \le C \int_J \|\partial_{ν_g}u(x,t)\|^2 \,\mathrm{d}S_g\,\mathrm{d}t . \] where $J\subset\partial\mathcal{M}\times(0,T)$ is measurable and has positive surface--time measure. A localized theorem requiring $W^{2,\infty}$ regularity only near the observed boundary patch is retained as a separate consequence. The proof of the observability inequality combines quantitative continuation from positive-measure boundary sets with low-frequency spectral concentration. This result shows that boundary observability of solutions to the heat equation can also be achieved in $W^{2,\infty}$ domains. Finally, as an application, we prove that measurements of the boundary normal derivative on $J$ uniquely determine the initial state of heat equation. Under an a priori bound in $D(-Δ_{g, D}^{s/2})$, the recovery satisfies a logarithmic stability estimate of optimal order $s/2$.
CommentsIn this version, we extend the results of the previous submission to the $W^{2,\infty}$-manifold case and discuss some of their applications in inverse problems
