ASYMPTOTIC BEHAVIOR OF GREEN'S FUNCTIONS FOR PARABOLIC AND ELLIPTIC EQUATIONS WITH CONSTANT COEFFICIENTS
M A Evgrafov, Mikhail Mikhailovich Postnikov · Mathematics of the USSR-Sbornik · 1970
The form of order , which is a function of the variables , where and is called strongly convex if the quadratic form (in a space of dimension equal to the number of the multi-indices with ) is positive definite. All even-order differentials of a strongly convex form are positive definite forms.The paper considers the parabolic equation , with a characteristic form which is strongly convex, and the asymptotic behavior of its Green's function for is derived. It is an unexpected property that this asymptotic behavior is dependent not on all saddle points of the corresponding integral with , but only on some of these. (This effect has not been observed for the previously known cases, with or .)The asymptotic behavior of the Green's function (for ) is derived also for the corresponding elliptic equation . It is suggested that analogous results hold for all convex forms , i.e. all forms having a positive definite second differential.Bibliography: 4 items.