This article is cited in
21 papers
Asymptotic behavior of Green's functions for parabolic and elliptic equations with constant coefficients
M. A. Evgrafov,
M. M. Postnikov
Abstract:
The form $P(\xi)=\sum_{|\mathfrak p|=2m}a_\mathfrak p\binom{2m}{\mathfrak p}\xi^\mathfrak p$ of order
$2m>0$, which is a function of the
$n$ variables
$\xi_1,\dots,\xi_n$, where
$\mathfrak p=(p_1,\dots,p_n)$,
$|\mathfrak p|=p_1+\dots+p_n$,
$\xi^\mathfrak p=\xi_1^{p_1}\cdots\xi_n^{p_n}$ and $\binom{2m}{\mathfrak p}=\frac{(2m)!}{p_1!\cdots p_n!}$, is called strongly convex if the quadratic form
$\sum_{|\mathfrak m|=|\mathfrak n|=m}a_{\mathfrak m+\mathfrak n}\mathrm X_\mathfrak m\mathrm X_\mathfrak n$
(in a space of dimension equal to the number of the multi-indices
$\mathfrak m$ with
$|\mathfrak m|=m$) is positive definite. All even-order differentials of a strongly convex form are positive definite forms.
The paper considers the parabolic equation $\frac{\partial u}{\partial t}+P\bigl(\frac1i\frac\partial{\partial x}\bigr)u=0$, with a characteristic form
$P(\xi)$ which is strongly convex, and the asymptotic behavior of its Green's function for
$t\to+0$ is derived. It is an unexpected property that this asymptotic behavior is dependent not on all saddle points of the corresponding integral with
$\operatorname{Re}P<0$, but only on some of these. (This effect has not been observed for the previously known cases, with
$n=1$ or
$m=1$.)
The asymptotic behavior of the Green's function (for
$\lambda\to+\infty$) is derived also for the corresponding elliptic equation $P\bigl(\frac1i\frac\partial{\partial x}\bigr)u+\lambda u=0$. It is suggested that analogous results hold for all convex forms
$P(\xi)$, i.e. all forms having a positive definite second differential.
Bibliography: 4 titles.
UDC:
517.947
MSC: 34B27,
35B40,
35K10,
35K65,
35J15,
35J70 Received: 11.12.1969