1,720,981 research outputs found
Ipoellitticità e buona posizione del problema di Cauchy per una classe di operatori ellittici degeneri
No full textA problem of transversal anisotropic ellipticity
No full textIn this paper we give an invariant approach in the spirit of Boutet-
Grigis-Helffer for a pseudodifferential operator with Characteristic Manifold given by a symplectic cone of codimension 2 v , which
contains the Grushin model
On the spectrum of an anharmonic oscillator
No full textWe consider the one-dimensional anharmonic oscillator P and we
study the qualitative behaviour of its eigenvalues; in particular, we show how the sign of its eigenvalues depends on the coefficients of P. We applied our results to the study of hypoellipticity and of a priori estimates for certain non-transversally elliptic operators
Analytic Hypoellipticity and the Treves Conjecture
Full text linkWe are concerned with the problem of the analytic hypoellipticity; precisely, we focus on the real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We discuss a model operator, P, (firstly appeared and studied in [3]) having a single symplectic stratum and prove that it is not analytic hypoelliptic. This yields a counterexample to the sufficient part of Treves conjecture; the necessary part is still an open problem
Analytic regularity for solutions to sums of squares: an assessment
Full text linkWe present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions
to sums of squares of vector fields satisfying the Hörmander condition
On a new method of proving Gevrey hypoellipticity for certain sums of squares
No full textWe consider an operator being a sum of squares of vector fields. It has the form, p,r∈N, P(x,Dx,Dy,Dt)=Dx2+x2(p-1)(Dy-xrDt)2. This type of operator is C∞ hypoelliptic by Hörmander's theorem, [18]. Its analytic or Gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the Treves conjecture. The Poisson-Treves stratification of P includes both symplectic and non-symplectic strata.In this paper we show that P is Gevrey (p+. r)/. p hypoelliptic, by constructing a parametrix whose symbol belongs to some exotic classes. One can also show that this number is optimal
On a Model “Sum of Squares” Operator
Full text linkWe study the real analytic and Gevrey regularity of the solutions to a type of “sum of squares” model operator, see (1), in two variables and obtain a result in agreement with Treves conjecture
On the analytic singular support for the solutions of a class of degenerate elliptic operators
Full text linkWe study a class of degenerate elliptic operators (which is a slight extension of the sums of squares of real-analytic vector fields satisfying the Hörmander condition). We show that, in dimensions 2 and 3, for every operator L in such a class and for every distribution u such that Lu is real-analytic, the analytic singular support of u, singsuppu, is a “negligible” set. In particular, we provide (optimal) upper estimates for the Hausdorff dimension of singsuppu. Finally, we show that in dimension n≥4, there exists an operator in such a class and a distribution u such that singsuppu is of dimension n
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