Heat Equation - Mean-value Property For The Heat Equation

Mean-value Property For The Heat Equation

Solutions of the heat equations

satisfy a mean-value property analogous to the mean-value properties of harmonic functions, solutions of

,

though a bit more complicated. Precisely, if u solves

and

then

where E_λ is a "heat-ball", that is a super-level set of the fundamental solution of the heat equation:

Notice that

as λ → ∞ so the above formula holds for any (x, t) in the (open) set dom(u) for λ large enough. Conversely, any function u satisfying the above mean-value property on an open domain of Rn × R is a solution of the heat equation. This can be shown by an argument similar to the analogous one for harmonic functions.