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:
“All over this land women have no political existence. Laws pass over our heads that we can not unmake. Our property is taken from us without our consent. The babes we bear in anguish and carry in our arms are not ours.”
—Lucy Stone (1818–1893)
“The LORD will afflict you with consumption, fever, inflammation, with fiery heat and drought, and with blight and mildew...”
—Bible: Hebrew, Deuteronomy 28:22.
“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)