HYPOELLIPTIC 2ND ORDER DIFFERENTIAL EQUATIONS
HYPOELLIPTIC 2ND ORDER DIFFERENTIAL EQUATIONS
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DOI：
10.1007/bf02392081
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发表时间：
1967-01-01
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影响因子：
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通讯作者：
HORMANDE.L
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文献类型：
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作者：
HORMANDE.L
在开集~ cRn（或流形）中具有Cr系数的线性微分算子P称为亚椭圆，如果对~ cRn中的每个分布u，我们有singsuppu = singsuppPu，即如果u在Pu为C函数的每个开集中必为C函数.当系数为常数时，P为亚椭圆的充要条件已经知道了很长时间（见[3]，第11章）。（第四节）。还表明，此类方程在受到变系数“较弱”算子的扰动后仍保持亚椭圆性（见[3，第3章）。VIII）。利用伪微分算子，可以进一步扩展容许扰动类;特别是，可以用这种方法得到许多类在变量变化下不变的亚椭圆（微分）方程（见[2]）。粗略地说，文[2]中给出的次椭圆性的充分条件是指将系数中的自变量”冻结”在某一点x上而得到的常系数微分方程是次椭圆的，且不随x变化太快。然而，文[2]给出的亚椭圆性的充分条件远非必要。例如，方程~ 2 u ~-x Au~ x y~ t（1.1）不满足它们，因为通过冻结一点上的系数而得到的算子必须只沿沿着二维平面运动，所以它不能是亚椭圆。但是Kolmogorov [8]在1934年已经构造了（1.1）的一个显式基本解，它是对角线外的一个C162函数，这意味着（1.1）是亚椭圆的。
A linear differential operator P with C r coefficients in an open set~ c R n (or a manifold) is called hypoelliptie if for every distribution u in~ we have sing supp u= sing supp Pu, that is, if u must be a C function in every open set where Pu is a C function. Necessary and sufficient conditions for P to be hypoelliptic have been known for quite some time when the coefficients are constant (see [3, Chap. IV]). It has also been shown that such equations remain hypoelliptic after a perturbation by a" weaker" operator with variable coefficients (see [3, Chap. VIII). Using pseudo-differential operators one can extend the class of admissible perturbations further; in particular one can obtain in that way many classes of hypoelliptic (differential) equations which are invariant under a change of variables (see [2]). Roughly speaking the sufficient condition for hypoelliptieity given in [2] means that the differential equations with constant coefficients obtained by" freezing" the arguments in the coefficients at a point x shall be hypoelliptie and not vary too rapidly with x. However, the sufficient conditions for hypoelliptieity given in [2] are far from being necessary. For example, they are not satisfied by the equation~ 2u~-x au~ x y~ t it'(1.1) for the operator obtained by freezing the coefficients at a point must operate along a two dimensional plane only so it cannot be hypoelliptie. But Kolmogorov [8] constructed already in 1934 an explicit fundamental solution of (1.1) which is a C 162 function outside the diagonal, and this implies that (1.1) is hypoelliptic.