Global s-solvability, global s-hypoellipticity and Diophantine phenomena

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.     

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.     

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     

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)     

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     

Regularity of Traveling Free Surface Water Waves with Vorticity

We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function of class C ^2,μ admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals 1, then we obtain analyticity of the stream function. The regularity results hold not only for periodic or solitary-water waves, but also for any solution to the hydrodynamic equations of class C ^2,μ .     

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.     

FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand [10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch [1]), improving the constantC _0,∞ of the results of Hargé-Lebeau [13] and Sjöstrand-Zworski [13] valid in C^∞ The works [3], [13] and [10] feature an adapted complex scaling for convex obstacles, but in [10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.     

Global hypoellipticity and 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 that if P is s-globally hypoelliptic in Tⁿ then its transposed operator tP is s-globally solvable in Tⁿ, thus extending to the Gevrey classes the well-known analogous result in the corresponding C^∞ class.     

Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear pdes

Given a Gs-involutive structure, (M,V), a Gevrey submanifold X⊂M which is maximally real and a Gevrey function u₀ on X we construct a Gevrey function u which extends u₀ and is a Gevrey approximate solution for V. We then use our construction to study Gevrey micro-local regularity of solutions, u∈C²(RN), of a system of nonlinear pdes of the form

     

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.     

Gevrey vectors in involutive tube structures and Gevrey regularity for the solutions of certain classes of semilinear systems

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.     

Global regularity in ultradifferentiable classes

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.     

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 Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

Hypoellipticity and Local Solvability in Gevrey Classes

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s. We prove first that if P is s‐hypoelliptic then its transposed operator ^tP is s‐locally solvable, thus extending to the Gevrey classes the well‐known analogous result in the C^∞class. We prove also that if P is s‐hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s‐hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.     

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

Local solvability for partial differential equations with multiple characteristics in mixed Gevrey-C ∞ spaces

In this paper, we consider a class of linear partial differential equations with multiple characteristics, whose principal part is elliptic in a set of variables. We assume that the subprincipal symbol has real part different from zero and that its imaginary part does not change sign. We then prove the local solvability of such a class of operators in mixed Gevrey-C ^∞ spaces, in the sense that the linear equation admits a local solution when the datum is Gevrey in some variables and only C ^∞ in the other ones.     

Strichartz estimates for dispersive equations and solvability of the Kawahara equation

We first establish a series of Strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by Kenig, Ponce and Vega. Next we use such estimates to study solvability of the Cauchy problem of the Kawahara equation in the class C(R,Hs(R)). Local existence is proved for s>1/4 and global existence is proved for s?2.     

Spectral problems in Lipschitz domains

The paper is devoted to spectral problems for strongly elliptic second-order systems in bounded Lipschitz domains. We consider the spectral Dirichlet and Neumann problems and three problems with spectral parameter in conditions at the boundary: the Poincaré–Steklov problem and two transmission problems. In the style of a survey, we discuss the main properties of these problems, both self-adjoint and non-self-adjoint. As a preliminary, we explain several facts of the general theory of the main boundary value problems in Lipschitz domains. The original definitions are variational. The use of the boundary potentials is based on results on the unique solvability of the Dirichlet and Neumann problems. In the main part of the paper, we use the simplest Hilbert L ₂-spaces H _s , but we describe some generalizations to Banach spaces H ^s _p of Bessel potentials and Besov spaces B ^s _p at the end of the paper.