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:
“General education is the best preventive of the evils now most dreaded. In the civilized countries of the world, the question is how to distribute most generally and equally the property of the world. As a rule, where education is most general the distribution of property is most general.... As knowledge spreads, wealth spreads. To diffuse knowledge is to diffuse wealth. To give all an equal chance to acquire knowledge is the best and surest way to give all an equal chance to acquire property.”
—Rutherford Birchard Hayes (1822–1893)
“For my part, when I enter most intimately into what I call myself, I always stumble on some particular perception or other, of heat or cold, light or shade, love or hatred, pain or pleasure. I never can catch myself at any time without a perception, and never can observe anything but the perception.”
—David Hume (1711–1776)
“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)