Asymptotic behavior of Green's functions for parabolic and elliptic equations with constant coefficients

M. A. Evgrafov ; M. M. Postnikov

Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 1-24

Résumé

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 $\operatorname{Re}P0$, 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.