Sharp Continuity Moduli for Dirichlet Heat Flow in Boundary-Reservoir Transport Metrics

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 $\{\mu :\mu (\mathrm{\Omega })\le m\}$, it is $1\mathrm{/}p$-Hölder: $$W_{b,p}(P_t\mu ,P_t\nu {)}^p\le C_{t,p,m,\mathrm{\Omega }}{W}_{b,p}(\mu ,\nu )\mathrm{.}$$ For $p>1$, we show that the exponent $1\mathrm{/}p$ 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 $\epsilon$ from $\mathrm{\partial }\mathrm{\Omega }$ has input $W_{b,p}$-distance $O(\epsilon )$ from zero, whereas after any fixed positive time, its $p$-th boundary moment is bounded below by $c\epsilon$. As a result, in the quadratic case and in the original finite-measure $W_{b,2}$ metric, there does not exist a standard finite-$\lambda$ ${\mathrm{E}\mathrm{V}\mathrm{I}}_{\lambda }$ 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.