Hypoelliptic second order differential equations

@article{Hrmander1967HypoellipticSO,
  title={Hypoelliptic second order differential equations},
  author={Lars H{\"o}rmander},
  journal={Acta Mathematica},
  year={1967},
  volume={119},
  pages={147-171}
}
  • L. Hörmander
  • Published 1 December 1967
  • Mathematics
  • Acta Mathematica
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]). I t has also been shown tha t 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… 
Hypoellipticity in the infinitely degenerate regime
Let {Xj} be a collection of real vector fields with C∞ coefficients, defined in a neighborhood of a point x0 ∈ R. Consider a second order differential operator L = − ∑ j X 2 j + ∑ j αjXj + β where
Hypoellipticity for a class of the second order partial differential equations
In this paper, we shall investigate the hypoellipticity for a class of degenerate equations of the second order with complex coefficients as a direct extension of the results obtained in [8]. As is
Hypoellipticity for a class of degenerate elliptic operators of second order
for Xi^O. Hormander's results in [2] can not be applicable to L when (xi) has a zero of infinite order. Compared with higher dimensional cases, the problem in R2 becomes much simpler. So one can
Failure of analytic hypoellipticity in a class of differential operators
For the hypoelliptic differential operators P = ∂2 x + ( x∂y − x∂t )2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P
Hypoellipticity for infinitely degenerate elliptic operators
Introduction. In the recent paper [5] Kusuoka-Strook gave a sufficient condition of hypoellipticity for degenerate elliptic operators of second order, as an application of the Malliavin calculus (see
A Necessary Condition For Analytic Hypoellipticity
A partial differential operator, L, is said to be analytic hypoelliptic in some open set Ω, if for every open U ⊂ Ω and every distribution u ∈ D′(U), if Lu ∈ C(U), then u ∈ C(U). Let X1, X2 be real
SOME ASPECTS OF DIFFERENTIAL GEOMETRY ASSOCIATED WITH HYPOELLIPTIC SECOND ORDER OPERATORS
I investigate some aspects of the geometry of the ^characteristics of a class of hypoelliptic second order partial differential operators. The resulting geometry looks quite a bit like Riemannian
Applications of analysis on nilpotent groups to partial differential equations
The last few years have witnessed the birth of a body of techniques for obtaining refined regularity theorems for certain hypoelliptic differential operators through analysis of homogeneous
Subelliptic estimates for some systems of complex vector fields: Quasihomogeneous case
For about twenty five years it was a kind of folk theorem that complex vector-fields defined on Ω x R t (with Ω open set in R n ) by L j = ∂ ∂t j + i∂φ ∂t j (t)∂ ∂x, j=1,...,n, t ∈ Ω, x ∈R, with φ
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 12 REFERENCES
Boundaries of Complex Manifolds
If M is a component of the boundary of a complex n-dimensional manifold X, then M has real dimension 2n - 1 and at each point x ∈ M the complexified tangent space T x has a distinguished (n -
Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung)
In zwei friuheren Arbeiten' (im folgenden als I und II zitiert) habe ich eine allgemeine Theorie der stetigen zufalligen Prozesse entwickelt. Es wurde dort unter sehr allgemeinen Voraussetzungen
Non‐coercive boundary value problems
Pseudo-differential operators and hypoelliptic equations
  • To appear in Amer. Math. Soc. Proc. Symp. Pure Math. ,
  • 1967
Linear second order equations with non-negative characteristic form
  • Russian.) Mat. Sb
  • 1966
The structure o / L i e groups
  • Holden -Day Inc.,
  • 1965
On a class of ultraparabolie equations
  • Russian.) Doklady Akad. NauIc SSSR
  • 1964
...
1
2
...