Subparabolic function

subcaloric function

The analogue of a subharmonic function for the heat equation

$$\tag{* } \frac{\partial u }{\partial t } - \Delta ^ {2} u = 0,$$

where $u= u( x, t)$, $x = ( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n}$ and $\Delta ^ {2} u = \sum _ {j=1} ^ {n} \partial ^ {2} u / \partial x _ {j} ^ {2}$ is the Laplace operator. For example, a function $v = v( x, t)$, $x \in \mathbf R$, $t > 0$, of class $C ^ {2}$ will be a subparabolic function in the rectangle

$$D = \{ {( x, t) \in \mathbf R \times \mathbf R _ {+} } : {a < x < b, 0 < t < h } \}$$

if

$$\frac{\partial v }{\partial t } - \frac{\partial ^ {2} v }{\partial x ^ {2} } \leq 0$$

everywhere in $D$. In a more general case, let the point $( x _ {0} , t _ {0} ) \in D$, let $\Delta$ be a sufficiently small equilateral triangle with base parallel to the axis $t= 0$ and let $( x _ {0} , t _ {0} ) \in \Delta \subset D$. A function $v = v( x, t)$ that is continuous in the closed domain $\overline{D}\;$ is said to be subparabolic in $D$ if its value at any point $( x _ {0} , t _ {0} ) \in D$ is not greater than the value at this point of that solution of (*) in any sufficiently small triangle $\Delta$, $( x _ {0} , t _ {0} ) \in \Delta$, that has the same values on the sides of $\Delta$ as $v( x, t)$.

Many properties of subharmonic functions, including the maximum principle, are also valid for subparabolic functions.

References

[1]	V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian)
[2]	I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)
[3]	I.G. Petrovskii, "Zur ersten Randwertaufgabe der Wärmeleitungsgleichung" Compos. Math. , 1 (1935) pp. 383–419

Comments

See [a1] for an account of subparabolic functions from a potential-theoretic point of view.

References

[a1]	J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1983)
[a2]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)