Separating inside versus outside translation directions near non-regular boundaries

Determine a rigorous criterion or method, for domains Ω with non-regular boundary ∂Ω, that separates translation directions v ∈ R^n into “inside” versus “outside” directions in the analysis of the set Ω_{η(t)} \ Ω_{√(4Dt)|v|} used for short-time heat content asymptotics; specifically, characterize when points x in Ω_{√(4Dt)|v|} are moved by x + 2√(Dt) v into Ω_{η(t)} so that χ_{(v,t)}(x) vanishes.

Background

In the derivation of bounds for heat content in the constant-coefficient case, the indicator χ{(v,t)}(x) depends on whether a translated point x + 2√(Dt) v remains in the interior Minkowski sausage Ω{η(t)}. For regular boundaries the geometry is controllable, but for non-regular (fractal or rough) boundaries the separation of translation directions v that keep points inside versus outside is nontrivial.

The authors explicitly note that this separation is unresolved for non-regular boundaries, and resolving it would improve the precision of short-time heat content estimates by identifying admissible translation directions relative to the boundary geometry.