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Mathematics of the USSR-Sbornik, 1970, Volume 11, Issue 1, Pages 1–24
DOI: https://doi.org/10.1070/SM1970v011n01ABEH002060
(Mi sm3432)
 

This article is cited in 21 scientific papers (total in 21 papers)

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

M. A. Evgrafov, M. M. Postnikov
References:
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.
Received: 11.12.1969
Bibliographic databases:
UDC: 517.947
Language: English
Original paper language: Russian
Citation: M. A. Evgrafov, M. M. Postnikov, “Asymptotic behavior of Green's functions for parabolic and elliptic equations with constant coefficients”, Math. USSR-Sb., 11:1 (1970), 1–24
Citation in format AMSBIB
\Bibitem{EvgPos70}
\by M.~A.~Evgrafov, M.~M.~Postnikov
\paper Asymptotic behavior of Green's functions for parabolic and elliptic equations with constant coefficients
\jour Math. USSR-Sb.
\yr 1970
\vol 11
\issue 1
\pages 1--24
\mathnet{http://mi.mathnet.ru//eng/sm3432}
\crossref{https://doi.org/10.1070/SM1970v011n01ABEH002060}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=273206}
\zmath{https://zbmath.org/?q=an:0233.35011}
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  • https://doi.org/10.1070/SM1970v011n01ABEH002060
  • https://www.mathnet.ru/eng/sm/v124/i1/p3
  • This publication is cited in the following 21 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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