Gevrey global solvability of non-singular real first-order differential operators

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.     

Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators

Abstract In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99     

Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions

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     

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

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. 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.     

On Gevrey solvability and regularity

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)     

Regularity and solvability of linear differential operators in Gevrey spaces

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.     

Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions

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.

     

Global solvability on the torus for certain classes of formally self-adjoint operators

In this paper we prove a necessary and sufficient condition for global solvability on the torus for two classes of formally self-adjoint operators. For the first class of operators we prove that global solvability is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For the second class of operators, when the coefficients are not identically zero, an independence condition on the coefficients is shown to be necessary and sufficient for global solvability. Received: 21 June 1999 / Revised version: 8 May 2000     

Sur un théorème de Cauchy–Kowalewski–Nagumo globalOn a global Cauchy–Kowalewski–Nagumo theorem

In this Note, we solve some global linear Cauchy problems. In the holomorphic case, for some operators, our result extends those of [4] when the initial data is in Gevrey class. In the continuous case, we give a generalisation of the local result of N. Nagumo, and on other hand, our result is an extension of the results of [5,1] for Gevrey initial data without the hyperbolicity condition. To cite this article: D. Gourdin, M. Mechab, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 563–567.     

Gevrey local solvability in locally integrable structures

We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex \(\mathrm{d}^{\prime }\) naturally induced by the de Rham complex. We prove that the so-called condition \(Y(q)\) on the signature of the Levi form, for local solvability of \(\mathrm{d}^{\prime }u=f\) , is still necessary even if we take \(f\) in the classes of Gevrey functions and look for solutions \(u\) in the corresponding spaces of ultradistributions.     

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Ω?=(−?,?)×S¹, given by L=∂/∂t+(a(x)+ib(x))∂/∂x, b?0, near the characteristic set Σ={0}×S¹. We show that the interplay between the order of vanishing of the functions a and b at x=0 plays a role in the Gevrey solvability.     

The cauchy problem for a class of evolution systems of the parabolic type with unbounded coefficients

For a class of evolution systems of the parabolic type with unbounded coefficients, we study the properties of the fundamental solution matrices and establish the well-posed solvability of the Cauchy problem for these systems in spaces of distributions similar to Gevrey ultradistributions. For a subclass of such systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Cauchy problem for parabolic systems with convolution operators in periodic spaces

For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{ \vec p,\vec h\} $ -parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

The Dirichlet and Regularity Problems for Some Second Order Linear Elliptic Systems on Bounded Lipschitz Domains

In this paper, we investigate divergence-form linear elliptic systems on bounded Lipschitz domains in \(\mathbb {R}^{d+1}, d \ge 2\), with L² boundary data. The coefficients are assumed to be real, bounded, and measurable. We show that when the coefficients are small, in Carleson norm, compared to one that is continuous on the boundary, we obtain solvability for both the Dirichlet and regularity boundary value problems given that the coefficients satisfy a certain “pseudo-symmetry” condition.     

Global solvability of real analytic complex vector fields in two variables

This paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables. The vector field is assumed to satisfy the Nirenberg-Treves condition (P) for local solvability. Normal forms for the vector field near the one-dimensional orbits are obtained and a generalization of the Riemann-Hilbert problem is considered.     

Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function

In this paper we consider the discrete one-dimensional Schrödinger operator with quasi-periodic potential v_n=λv(x+nω). We assume that the frequency ω satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove—in the perturbative regime—that for large disorder λ and for most frequencies ω the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.     

Propagation des singularités Gevrey pour l’équation de Cauchy-Riemann dégénérée

In this paper we study the degenerate Cauchy-Riemann equation in Gevrey classes. We first prove the local solvability in Gevrey classes of functions and ultra-distributions. Using microlocal techniques with Fourier integral operators of infinite order and microlocal energy estimates, we prove a result of propagation of singularities along one dimensional bicharacteristics.      

The problem of determining the one-dimensional kernel of the electroviscoelasticity equation

We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.     

Boundary value problems for a higher-order parabolic equation with growing coefficients

We establish the unique solvability of boundary value problems in Hölder function classes for a linear parabolic equation of order 2m in noncylindrical domains of the class C ^{2m ? 1,α} , possibly unbounded (with respect to x as well as t), with nonsmooth (with respect to t) lateral boundary under the condition that the lower-order coefficients and the right-hand side of the equation can grow in a certain way when approaching the parabolic boundary of the domain and the leading coefficients may fail to satisfy the Dini condition near this boundary.     

On and Gevrey regularity of sublaplacians

In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a general class of sublaplacians when the finite type condition holds.