共查询到20条相似文献,搜索用时 62 毫秒
1.
In this paper we consider the problem of global analytic and Gevrey solvability for a class of partial differential operators
on a torus in the form of squares of vector fields. We prove that global analytic and Gevrey solvability on the torus is equivalent
to certain Diophantine approximation properties.
Mathematics Subject Classification (2000) 35D05, 46E10, 46F05, 58J99 相似文献
2.
Gerson Petronilho 《Indagationes Mathematicae》2005,16(1):67-90
In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class Gs(TN). For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition. 相似文献
3.
A. Alexandrou Himonas 《Proceedings of the American Mathematical Society》2001,129(7):2061-2067
In this paper we consider the problem of global Gevrey and analytic regularity for a class of partial differential operators on a torus in the form of a sum of squares of vector fields, which may not satisfy the bracket condition. We show that these operators are globally Gevrey or analytic hypoelliptic on the torus if and only if the coefficients satisfy certain Diophantine approximation properties.
4.
We obtain a global version in the N-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context. 相似文献
5.
《Mathematische Nachrichten》2018,291(5-6):729-758
We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral. 相似文献
6.
Existence and Regularity of Periodic Solutions to Certain First-Order Partial Differential Equations
We present conditions on the coefficients of a class of vector fields on the torus which yield a characterization of global solvability as well as global hypoellipticity, in other words, the existence and regularity of periodic solutions. Diophantine conditions and connectedness of certain sublevel sets appear in a natural way in our results. 相似文献
7.
Gerson Petronilho 《Mathematische Nachrichten》2009,282(3):470-481
In this paper we study global C∞ and Gevrey solvability for a class of sublaplacian defined on the torus T 3. We also prove Gevrey regularity for a class of solutions of certain operators that are globally C∞ hypoelliptic in the N ‐dimensional torus (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
8.
In this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus. This condition says that either a Diophantine condition is satisfied or there exists a point of finite type. Also, we describe the analytic and Gevrey versions of this result. The proof is based on L2-estimates and microlocal analysis. 相似文献
9.
Michael Christ 《偏微分方程通讯》2013,38(3-4):117-204
Hypoellipticity in Gevrey classes Gs is characterized for a simple family of sums of suares of verctor fields satisfying the bracket hypothesis, with analytic coefficients. It is shown that hypoellipticity holds if and only if s is greater than or equal to an optimal exponent that may take on any rational value. 相似文献
10.
Adalberto P. Bergamasco Sé rgio Luí s Zani 《Proceedings of the American Mathematical Society》2008,136(4):1305-1310
We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold.
In particular, we show that such vector fields exist only on the torus.
11.
Paulo D. Cordaro A. Alexandrou Himonas 《Transactions of the American Mathematical Society》1998,350(12):4993-5001
We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.
12.
Jairo E. Castellanos Paulo D. Cordaro Gerson Petronilho 《Journal d'Analyse Mathématique》2013,119(1):333-364
In this work, we introduce the notion of s-Gevrey vectors in locally integrable structures of tube type. Under the hypothesis of analytic hypoellipticity, we study the Gevrey regularity of such vectors and also show how this notion can be applied to the study of the Gevrey regularity of solutions of certain classes of semilinear equations. 相似文献
13.
Todor Gramchev Petar Popivanov Massafumi Yoshino 《Annali di Matematica Pura ed Applicata》1996,170(1):103-131
Summary
In this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.Partially supported by GNAFA, CNR, Italy.Partially supported by JSPS, Japan and a grant MM-410/94 with MES, Bulgaria.Partially supported by Chuo University special research fund. 相似文献
14.
We consider an operator P which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold,
but the rank of the symplectic form σ is not constant on Char P. Moreover the Hamilton foliation of the non-symplectic stratum of the Poisson-Treves stratification for P consists of closed curves in a ring-shaped open set around the origin. We prove that then P is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for P is G
k+1 and that this is sharp.
相似文献
15.
Paulo L. Dattori da Silva Maurício Fronza da Silva 《Annali di Matematica Pura ed Applicata》2013,192(2):245-253
In this article we deal with Gevrey global solvability of non-singular first-order operators defined on an n-dimensional s-Gevrey manifold, s > 1. As done by Duistermaat and Hörmander in the C ∞ framework, we show that Gevrey global solvability is equivalent the existence of a global cross section. 相似文献
16.
We study ω-regularity of the solutions of certain operators that are globally C ∞-hypoelliptic in the N-dimensional torus. We also apply these results to prove the global ω-regularity for some classes of sublaplacians. In this way, we extend previous work in the setting of analytic and Gevrey classes. Different examples on local and global ω-hypoellipticity are also given. 相似文献
17.
Adalberto P. Bergamasco 《Transactions of the American Mathematical Society》1999,351(10):4113-4126
We present a characterization of the operators
which are globally analytic hypoelliptic on the torus. We give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares.
18.
A necessary and sufficient condition is given for a sum of squares operator to be globally hypoelliptic on an N-dimensional torus. This condition is expressed in terms of Diophantine approximation properties of the coefficients. The proof of the Theorem is based on L2-estimates and microlocalization. 相似文献
19.
Abed Bounemoura 《Regular and Chaotic Dynamics》2013,18(3):237-260
In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic Hamiltonians, and only in the periodic case for Gevrey Hamiltonians. As applications, we obtain an exponentially large upper bound on the stability time for the evolution of the action variables and an exponentially small upper bound on the splitting of invariant manifolds for hyperbolic tori, generalizing corresponding results for analytic Hamiltonians. 相似文献
20.
We consider real analytic involutive structures 𝒱, of co-rank one, defined on a real analytic paracompact orientable manifold M. To each such structure we associate certain connected subsets of M which we call the level sets of 𝒱. We prove that analytic regularity propagates along them. With a further assumption on the level sets of 𝒱 we characterize the global analytic hypoellipticity of a differential operator naturally associated to 𝒱. As an application we study a case of tube structures. 相似文献