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.     

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.     

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.     

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 .     

On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

We study the wellposedness in the Gevrey classes G^s and in C^∞ of the Cauchy problem for weakly hyperbolic equations of higher order. In this paper we shall give a new approach to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cauchy problem for equations with multiple characteristics

This paper contains a proof of γ^n(χ) correctness of the noncharacteristic Cauchy problem for nonstrictly hyperbolic equations with analytic coefficients under the condition that its characteristic roots are smooth and under some additional assumptions on the lower-order terms. There are two extreme cases: (1) $\text{χ <}\text{r}\text{r}\text{? 1}$. In this case condition (0.6) is “void,” and we do not require conditions on P_s for s < m. For this case, see [3, 8]. (2) Case of constant multiplicity of characteristic roots and χ = +∞. In this case condition (0.6) implies conditions on P_s, where s = m, m ? 1,…, m ? r + 1, i.e., up to the same order as the necessary condition for C^∞-correctness [2]. Recall that in the case of equations with characteristics of constant multiplicity condition (0.6) (Levi's condition in this case) for χ = ∞ is necessary [2, 4] and sufficient [1] for C^∞-correctness.     

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.     

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)     

On graphs with small subgraphs of large chromatic number

Bollobás, Erdös, Simonovits, and Szemerédi conjectured [1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn ² edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdös [2]: For each positive constantc and positive integerk there exist positive integersf _k(c) andn _o such that ifG is any graph with more thann _o vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn ² edges, thenG contains ak-chromatic subgraph with at mostf _k(c) vertices.     

Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations

We present a very simple proof of the global existence of a C^∞ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has C^∞ dependence on initial data u₀ in the class of H^s divergence-free vector fields for s > 2.     

Multipliers of the Hölder classes

Following the lines of [13] we introduce the classes of mixed smoothness $\text{L}$α_p(Rⁿ), $\text{L}$α_p(Zⁿ) for a multi-index α = (α₁,…, α_n). Such classes are naturally tied up with the study of semi-elliptic differential and difference equations.Besides a brief presentation of such classes, we concentrate our research on the study of mixed homogeneous multipliers with homogeneity β = (β₁, …, β_n) and their preservation of mixed homogeneous Hölder classes $\text{L}$_α^∞, for a different multi-index α.In the last paragraph we apply the results to produce various improvements of the classical Schauder's estimates, for differential and difference equations, in the parabolic and elliptic case.     

Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.     

Exotic characteristic classes of quaternionic bundles

For aC ^∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kähler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].     

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

     

Opérateurs d'extension linéaires explicites dans des intersections de classes ultradifférentiables

B. S. Mityagin proved that the Chebyshev polynomials form a Schauder basis of the space of C ^∞ functions on the interval [–1,1]. Whereof he deduced an explicit continuous linear extension operator. These results were extended, by A. Goncharov, to compact sets without Markov's property. On the reverse, M. Tidten gave examples of compact sets for which there is no continuous linear extension operator. In this paper, we generalize these works to the intersections of ultradifferentiable classes of functions built on the model of the non quasianalytic intersection of Gevrey classes. We get, among other things, a Whitney linear extension theorem for ultradifferentiable jets of Beurling type.     

Opérateurs linéaires continus d'extension dans des intersections de classes ultradifférentiables

Mityagin proved that the Tchebyshev polynomials form a Schauder basis of the space of C^∞ functions on the interval [?1,1]. Thus, he deduced an explicit continuous linear extension operator. These results were extended, by Goncharov, to compact sets which do not satisfy the Markov's inequalities. On the other hand, Tidten gave examples of compact sets for which there is no continuous linear extension operator. In this Note, we generalize these works to ultradifferentiable classes of functions built on the model of the intersection of non quasi-analytic Gevrey classes. We get, among other things, a Whitney linear extension theorem for ultradifferentiable jets of Beurling type. To cite this article: P. Beaugendre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On bijections of Lorentz manifolds,which leave the class of spacelike paths invariant

Suppose that (M, g) and (M′, g′) are Lorentz manifolds, and that f: M → M′ is a bijection, such that f and f^-1 preserve spacelike paths (f: M → M′ has this property, if for any spacelike path γ: J → M in (M ,g), the composition fγ: J → M′ is a spacelike path in (M′, g′)). Then f is a (manifold-) homeomorphism.This statement is the ‘spacelike’ version of an analogous ‘timelike’ theorem (Hawking, King and McCarthy [6] and Göbel [2] for strongly causal, and Malament [10] for general Lorentz manifolds).With this result it is possible to prove a conjecture of Göbel [3] which states that every bijection between time-orientable n-dimensional (n ? 3) Lorentz manifolds which preserves spacelike paths is a conformal C^∞-diffeomorphism.     

On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

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.     

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.