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:
“The rights and interests of the laboring man will be protected and cared for, not by the labor agitators, but by the Christian men to whom God in His infinite wisdom has given control of the property interests of the country.”
—George Baer (1842–1914)
“As in hoary winter’s night stood shivering in the snow,
Surprised I was with sudden heat which made my heart to glow;
And lifting up a fearful eye to view what fire was near,
A pretty Babe all burning bright did in the air appear;”
—Robert Southwell (1561?–1595)
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)