2026/07/18 by Maja Gwozdz
Mathematics · #math.AP
Let Ω⊂\mathbb Rn be a bounded C2 open set and let Pt be the killed Dirichlet heat semigroup. We prove the sharp fixed-time power-scale modulus of Pt for the Figalli--Gigli boundary-reservoir transport distances Wb,p. For every t>0, Pt is globally Lipschitz with respect to Wb,1. For every p>1, and on every total-mass sublevel \μ:μ(Ω)≤ m\, it is 1/p-Hölder: Wb,p(Ptμ,Ptν)p ≤ Ct,p,m,Ω Wb,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, Pt 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 ε from ∂Ω has input Wb,p-distance O(ε) from zero, whereas after any fixed positive time, its p-th boundary moment is bounded below by cε. As a result, in the quadratic case and in the original finite-measure Wb,2 metric, there does not exist a standard finite-λ EVIλ semigroup on a Wb,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.