Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group

We study the boundary values of the functions of the Sobolev function spaces W _∞ ^l and the Nikol’ski? function spaces H _∞ ^l which are defined on an arbitrary domain of a Carnot group. We obtain some reversible characteristics of the traces of the spaces under consideration on the boundary of the domain of definition and sufficient conditions for extension of the functions of these spaces outside the domain of definition. In some cases these sufficient conditions are necessary.     

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.     

Generalized Harmonic Functions and the Dewetting of Thin Films

This paper describes the solvability of Dirichlet problems for Laplace's equation when the boundary data is not smooth enough for the existence of a weak solution in H¹Ω. Scales of spaces of harmonic functions and of boundary traces are defined and the solutions are characterized as limits of classical harmonic functions in special norms. The generalized harmonic functions, and their norms, are defined using series expansions involving harmonic Steklov eigenfunctions on the domain. It is shown that the usual trace operator has a continuous extension to an isometric isomorphism of specific spaces. This provides a characterization of the generalized solutions of harmonic Dirichlet problems. Numerical simulations of a model problem are described. This problem is related to the dewetting of thin films and the associated phenomenology is described.     

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L _p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ?Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _p,σ (Ω), p ∈ (1,∞).     

Thin sequences in the corona of H ∞

In this paper we consider several conditions for sequences of points in M(H ^∞) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona $M(H^\infty )\backslash \mathbb{D}$ of H ^∞ is a thin sequence for H ^∞, which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H ^∞ necessarily satisfy the same condition.     

Approximation by herglotz wave functions

By a general argument, it is shown that Herglotz wave functions are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to the reduced wave equation in Ω. This is used to provide corresponding approximation results in global spaces (eg. in L2‐Sobolev‐spaces H^m(Ω)) and for boundary data. Copyright © 2004 John Wiley & Sons, Ltd.     

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).     

Boundary values of solutions of second-order linear parabolic equations

The local properties of the solutions of parabolic equations are investigated in domains whose boundaries are defined by the difference of two convex functions. Conditions for the existence of nontangential boundary and L₂ limits are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1433–1440, October, 1992.     

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L ^p and weak L ^p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L _Φ having the property L ^∞ ? L _Φ L ^p , 1 ? p > ∞. The second contains spaces L _Φ that resemble L ^p spaces.     

The structure of Banach algebras of bounded continuous functions on the open disk that contain H∞, the Hoffman algebra,and nontangential limits

LetH _B ^G be an algebra of bounded continuous functions on an open disk representable in the formH _B G, where andH _B is a closed subalgebra in C(D) consisting of the functions that have nontangential limits almost everywhere on {ie1023-06}, and these limits belong to the Douglas algebraB. In this paper we describe the spaceM(H _B ^G ) of maximal ideals of the algebraH _B ^G and prove thatM(H _B ^G ) =M(B) M(H _B ^G and prove thatM(H ₀ ^G ), whereH ₀ ^G is a closed ideal inG consisting of functions having nontangential limits equal to zero almost everywhere on {ie1023-12}. Moreover, it is established that on the disk. The Chang-Marshall theorem is generalized for the Banach algebrasH _B ^G . We also prove that for any Douglas algebraB, whereI B = {u } _B are inner functions such that on.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 924–931, July, 1993.     

FUNCTIONAL CALCULI FOR HILBERT SPACE OPERATORS

Abstract

Let T be a bounded operator on a Hilbert space H with Von Neumann spectral set X. If there exists no non-zero reducing subspace of H restricted to which T is a normal operator with spectrum contained in the boundary of X and if the uniform algebra R(X) is pointwise boundedly dense in H^∞ (X°), then there exists a functional calculus f → f(T) for f ε H^∞ (X°). A similar result for the two-variable case is also proved.     

One-sided maximal functions and Hp

Let N be the nontangential maximal function of a function u harmonic in the Euclidean half-space Rⁿ × (0, ∞) and let N^? be the nontangential maximal function of its negative part. If u(0, y) = o(y^?n) as y → ∞, then ∥N∥_p ? c_p ∥N^?∥_p, 0 < p < 1, and more. The basic inequality of the paper (Theor. 2.1) can be used not only to derive such global results but also may be used to study the behavior of u near the boundary. Similar results hold for martingales with continuous sample functions. In addition, Theorem 1.3 contains information about the zeros of u. For example, if u belongs to H^p for some 0 < p < 1, then every thick cone in the half-space must contain a zero of u.     

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λ_g: C^∞ (?M) → C^∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λ_gh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If g^t = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λ_g ^t = Λ_g + tΛ_? + o(t). We study the question: How do tensor fields ? look like for which Λ_? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.     

Hardy approximation toL ∞ functions on subsets of the circle

We consider approximation ofL ^∞ functions byH ^∞ functions on proper substs of the circle. We derive some properties of traces of Hardy classes on such subsets, and then turn to a generalization of classical extremal problems involving norm constraints on the complementary subset.     

On approximation of affine Baire-one functions

It is known (G. Choquet, G. Mokobodzki) that a Baire-one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire-one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces. In particular, we investigate the function spaceH(U), consisting of the functions continuous on the closure of a bounded open setU⊂ℝ ^m and harmonic onU, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the spaceB ₁ ^b (H(U)) of bounded functions which are pointwise limits of functions from the spaceH(U) and thatB ₁ ^b (H(U)) is not uniformly closed. On the other hand, every Baire-oneH(U)-affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging toH(U). It turns out that such a situation always occurs for simplicial spaces whereas it is not the case for general function spaces. The paper provides several characterizations of those Baire-one functions which can be approximated pointwise by bounded sequences of elements of a given function space. Research supported in part by grants GA ČR No. 201/00/0767 from the Grant Agency of the Czech Republic, GA UK 165/99 from the Grant Agency of Charles University, and in part by grant number MSM 113200007 from the Czech Ministry of Education.     

Existence and uniqueness of C0-semigroup in L∞: a new topological approachExistence et unicité de C0-semigroupe sur L∞

A sub-Markov semigroup in L^∞ is in general not strongly continuous with respect to the norm topology. We introduce a new topology on L^∞ for which the usual sub-Markov semigroups in the literature become C₀-semigroups. This is realized by a natural extension of the Phillips theorem about dual semigroup. A simplified Hille–Yosida theorem is furnished. Moreover this new topological approach will allow us to introduce the notion of L^∞-uniqueness of pre-generator. We present several important pre-generators for which we can prove their L^∞-uniqueness. To cite this article: L. Wu, Y. Zhang, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 699–704.     

Diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow with fixed viscosity

We establish magnetic diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed viscosity. The convergence is verified under various Sobolev norms, including the L~∞(L~2)and L~∞(H~1) norm.