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)