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 Eλ 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.
Read more about this topic: Heat Equation
Famous quotes containing the words property, heat and/or equation:
“Things have their laws, as well as men; and things refuse to be trifled with. Property will be protected.”
—Ralph Waldo Emerson (1803–1882)
“The Soul rules over matter. Matter may pass away like a mote in the sunbeam, may be absorbed into the immensity of God, as a mist is absorbed into the heat of the Sun—but the soul is the kingdom of God, the abode of love, of truth, of virtue.”
—Ralph Waldo Emerson (1803–1882)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)