Uniform bounds for the heat content of open sets in Euclidean space
  We obtain (i) lower and upper bounds for the heat content of an open set in $\mathbb{R}^m$ with $R$-smooth boundary and finite Lebesgue measure, (ii) a necessary and sufficient geometric condition for finiteness of the heat content in $\mathbb{R}^m$, and corresponding lower and upper bounds, (iii) lower and upper bounds for the heat loss of an open set in $\mathbb{R}^m$ with finite Lebesgue measure. 
Michiel van den Berg, Katie Gittins
math.AP