On the global solvability in Gevrey classes on the n-dimensional torus

Abstract

Let P be a linear partial differential operator with coefficients in the Gevrey class G s (T n ), where T n is the n-dimensional torus and s 1. We prove a necessary condition for the s-global solvability of P on T n . We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.  2004 Elsevier Inc. All rights reserved.In the last years many papers are concerned with the study of the global solvability and hypoellipticity of linear partial differential operators on compact manifolds, e.g., torus, in large scales of functional spaces (see, e.g., [3,[7][8][9][10][18][19][20]23] and references listed therein). It is well known that the theory of global properties of differential operators is not well developed in comparison with the one of local properties. In particular, the global properties are open problems except for certain classes of operators. On the other hand, the local and global solvability/hypoellipticity are rather different in general. In fact, there are (A.A. Albanese), popivano@banmatpc.math.bas.bg (P. Popivanov).

Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions

“…, n, are realvalued functions defined on T m . This result extends to analytic and Gevrey classes the one of Petronilho [11] where the C ∞ -global solvability for such a class of operators was characterized and to any dimensions of one of [14]. On the other hand, since there is a connection between the global analytic and Gevrey solvability of any partial differential operator P and the global analytic and Gevrey hypoellipticity of t P as it was shown in [15] (or see [16]), this paper can be regarded as a continuation of Himonas [6] where a study of the global analytic and Gevrey hypoellipticity for this class of operators was presented.…”

“…The main result of this paper is the following one which extends to Gevrey classes the results in [11] and to any dimension the ones of [14]. Its proof is along the lines of [11].…”

“…Consequently, as already observed in [14], a fundamental system of continuous seminorms for G s (T n ) can be defined as follows.…”

“…This definition allows, in principle, an infinite dimensional Ker t P (see [9] or Remark 3.2 in [14]). …”

“…Since the operator − t +w 2 (t, ξ) is elliptic with respect to t andĥ(t, ξ) ∈ G s (T m ) for all fixed ξ ∈ Z n , we also have thatû(t, ξ) ∈ G s (T m ). Then, by multiplying (14) byū and integrating by parts with respect to t ∈ T m , we obtain that, for all ξ ∈ A,…”