AI 中文总结
研究有界\(C^2\)开集\(\Omega\)中狄氏热半群\(P_t\)在边界-储层输运度量\(W_{b,p}\)下的连续性模，证明其固定时间幂尺度模，给出不同\(p\)取值下的性质及最优指数，还阐述了相关不连续情况与下界障碍。
AI 中文摘要
设\(\Omega\subset\mathbb R^n\)为有界\(C^2\)开集，\(P_t\)为被杀掉的狄氏热半群。我们证明了关于Figalli - Gigli边界-储层输运距离\(W_{b,p}\)的\(P_t\)的精确固定时间幂尺度模。对于每个\(t>0\)，\(P_t\)关于\(W_{b,1}\)是全局Lipschitz的。对于每个\(p>1\)，在每个总质量子水平\(\{\mu:\mu(\Omega)\le m\}\)上，它是\(1/p\)-Hölder的。对于\(p>1\)，我们表明幂模尺度中的指数\(1/p\)是最优的。在全有限测度空间上，\(P_t\)在零测度处不连续。为建立下界，我们依赖于边界层的放大。在二次情形和原始有限测度\(W_{b,2}\)度量中，不存在包含仿射常数边界数据类且能限制为仿射常数边界狄氏热流的\(W_{b,2}\)度量域上的标准有限\(\lambda\)\(\mathrm{EVI}_\lambda\)半群。最后，我们还描述了散度形式下光滑一致椭圆扰动的相应下界障碍。
英文摘要
Let $Ω\subset\mathbb R^n$ be a bounded $C^2$ open set and let $P_t$ be the killed Dirichlet heat semigroup. We prove the sharp fixed-time power-scale modulus of $P_t$ for the Figalli--Gigli boundary-reservoir transport distances $W_{b,p}$. For every $t>0$, $P_t$ is globally Lipschitz with respect to $W_{b,1}$. For every $p>1$, and on every total-mass sublevel $\{μ:μ(Ω)\le m\}$, it is $1/p$-Hölder: \[ W_{b,p}(P_tμ,P_tν)^p \le C_{t,p,m,Ω} W_{b,p}(μ,ν). \] For $p>1$, we show that the exponent $1/p$ is optimal in the scale of power moduli. On the full finite-measure space, $P_t$ is discontinuous at the zero measure. To establish the lower bound, we rely on the amplification of the boundary layer. More precisely, a unit mass initially placed at distance $\varepsilon$ from $\partialΩ$ has input $W_{b,p}$-distance $O(\varepsilon)$ from zero, whereas after any fixed positive time, its $p$-th boundary moment is bounded below by $c\varepsilon$. As a result, in the quadratic case and in the original finite-measure $W_{b,2}$ metric, there does not exist a standard finite-$λ$ $\mathrm{EVI}_λ$ semigroup on a $W_{b,2}$-metric domain which would contain the affine constant-boundary data class and could restrict to the affine constant-boundary Dirichlet heat flow. Finally, we also describe the corresponding lower-bound obstruction for smooth uniformly elliptic perturbations in divergence form.
