Remainder exponent β in the general asymptotic expansion for admissible domains

Ascertain the value and geometric dependence of the remainder exponent β in the asymptotic expansion N(t) = ∫_0^{+∞} (e^{−z^2}/√π) μ(∂Ω, √(4Dt) z) dz + o(t^β) as t → 0+ for admissible domains Ω ⊂ R^n (Equation (EqResVM)), including identification of bounds or exact expressions for β in terms of properties of ∂Ω.

Background

Proposition \ref{PropLimitCase} gives a general asymptotic form for heat content in terms of volumes of parallel sets, but introduces an unspecified remainder term o(t^β).

For domains with complex or fractal boundaries, specifying β would substantially refine the asymptotics. The authors note that the boundary’s shape is not explicitly known in the general setting, leaving β undetermined, and suggest numerical evidence may guide expectations.