Interpolation and harmonic majorants in big Hardy-Orlicz spaces

Free interpolation in Hardy spaces is characterized by the well-known Carleson condition. The result extends to Hardy-Orlicz spaces contained in the scale of classical Hardy spaces H ^p, p > 0. For the Smirnov and the Nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic majorants (quasi-bounded in the case of the Smirnov class). Since the Smirnov class can be regarded as the union over all Hardy-Orlicz spaces associated with a so-called strongly convex function, it is natural to ask how the condition changes from the Carleson condition in classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of this paper is to narrow down this gap from the Smirnov class to “big” Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences for a class of Hardy-Orlicz spaces that carry an algebraic structure and are strictly bigger than ⋃ _p>0 H ^p . It turns out that the interpolating sequences are again characterized by the existence of quasi-bounded majorants, but now the functions defining these quasi-bounded majorants have to be in suitable Orlicz spaces. The existence of harmonic majorants defined by functions in such Orlicz spaces is also discussed in the general situation. We finish the paper with a class of examples of separated Blaschke sequences which are interpolating for certain Hardy-Orlicz spaces without being interpolating for slightly smaller ones.     

Busemann Functions in a Harmonic Manifold

For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces.     

Integral representations of harmonic functions in half spaces

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u⁺(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u⁻(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.     

Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u ∈ H^1?k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation u_S ∈ S of u satisfies ‖u ? u_S‖ ≤ Ch^γ‖u‖, where h is the typical size of the “elements” defining the GFEM‐space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super‐approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H¹(??), we need also the duals of the Sobolev spaces H^m(??), m ∈ ?₊. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations

We investigate thc close relations existing between certain geometric properties of domains Ω of R^N, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation u_t-△u=|u|^p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in L^r(ω)1<r<∞ when r is supercritical(r>N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation u_t-△u=u^p-μ|u|^q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.     

A microscopic convexity principle for nonlinear partial differential equations

We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇ ² u is of constant rank for any convex solution u of equation F(∇ ² u,∇ u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed. Research of the first author was supported in part by NSFC No.10671144 and National Basic Research Program of China (2007CB814903). Research of the second author was supported in part by an NSERC Discovery Grant.     

Some nonconforming mixed box schemes for elliptic problems

In this article, we introduce three schemes for the Poisson problem in 2D on triangular meshes, generalizing the FVbox scheme introduced by Courbet and Croisille [1]. In this kind of scheme, the approximation is performed on the mixed form of the problem, but contrary to the standard mixed method, with a pair of trial spaces different from the pair of test spaces. The latter is made of Galerkin‐discontinuous spaces on a unique mesh. The first scheme uses as trial spaces the P¹ nonconforming space of Crouzeix‐Raviart both for u and for the flux p = ?u. In the two others, the quadratic nonconforming space of Fortin and Soulie is used. An important feature of all these schemes is that they are equivalent to a first scheme in u only and an explicit representation formula for the flux p = ?u. The numerical analysis of the schemes is performed using this property. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 355–373, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10003     

On point to point reflection of harmonic functions across real-analytic hypersurfaces in ℝ n

Let Γ be a non-singular real-analytic hypersurface in some domainU ⊂ ℝ ⁿ and let Har₀(U, Γ) denote the linear space of harmonic functions inU that vanish on Γ. We seek a condition onx ⁰,x ¹ ∈U/Γ such that the reflection law (RL)u(x ⁰)+Ku(x ¹)=0, ∀u∈Har₀(U, Γ) holds for some constantK. This is equivalent to the class Har₀ (U, Γ) not separating the pointsx ⁰,x ¹. We find that in odd-dimensional spaces (RL)never holds unless Γ is a sphere or a hyperplane, in which case there is a well known reflection generalizing the celebrated Schwarz reflection principle in two variables. In even-dimensional spaces the situation is different. We find a necessary and sufficient condition (denoted the SSR—strong Study reflection—condition), which we described both analytically and geometrically, for (RL) to hold. This extends and complements previous work by e.g. P.R. Garabedian, H. Lewy, D. Khavinson and H. S. Shapiro.     

On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ?³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?³ into a sum u = u⁺+u^? were u^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.     

On weighted inequalities for martingale transform operators

For 1 < p < ∞, the almost surely finiteness of is a necessary and sufficient condition in order to have almost surely convergence of the sequences {E(f|?_n)} with f ∈ L^p(v dP). This condition is also equivalent to have weighted inequalities from L^p(v dP) into L^p(u dP) for some weight u for Doob's maximal function, square function and generalized Burkholder martingale transforms. Similarly, E(u|?₁) < ∞ turns out to be necessary and sufficient for the above weighted inequalities to hold for some v.     

On the singular set of stationary harmonic maps

LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH ⁿ⁻² (Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW ^1,n (M, N), thenu is smooth. This extends results of F. Hélein for the casen=2, or the caseN is a sphere ([9], [10]).     

Two‐dimensional chemotaxis models with fractional diffusion

In this paper we study the Cauchy problem for the fractional diffusion equation u_t + (?Δ)^α/2u=?·(u?(Δ^?1u)), generalizing the Keller–Segel model of chemotaxis, for the initial data u₀ in critical Besov spaces ?(?²) with r∈[1, ∞], where 1<α<2. Making use of some estimates of the linear dissipative equation in the frame of mixed time–space spaces, the Chemin ‘mono‐norm method,’ Fourier localization technique and the Littlewood–Paley theory, we obtain a local well‐posedness result. We also consider analogous ‘doubly parabolic’ models. Copyright © 2008 John Wiley & Sons, Ltd.     

Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h ^l + k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.     

Function Spaces with Exponential Weights I

In this paper we define weighted function spaces of type B^s_pq(u) and F^s_pq(u) on the Euclidean space IRⁿ, where u is a weight function of at most exponential growth. In particular, u(x) = exp(±|χ|) is an admissible weight. We prove some basic properties of these spaces, such as completeness and density of the smooth functions.     

Sets of Harmonicity for Finely Harmonic Functions

Given an open set U in R ⁿ (n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.     

The Liouville property and a conjecture of De Giorgi

We consider bounded entire solutions of the nonlinear PDE Δu + u − u³ = 0 in ℝ^d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes. The proof uses a Liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator. © 2000 John Wiley & Sons, Inc.     

A perturbation theorem for operator semigroups in Hilbert spaces

Abstract. We prove a perturbation result for C ₀ -semigroups on Hilbert spaces and use it to show that certain operators of the form Au = i u ^(2k) +V· u ^(l) on L ² (R) generate a semigroup that is strongly continuous on (0,∞) .     

Stability of a transmission problem for Kirchhoff‐type wave equations with memory on the boundary

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff‐type wave equations with a memory condition on one part of the boundary. Without the condition u₀ = 0 on Γ₀, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd.     

Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity

In this short note we give a link between the regularity of the solution u to the 3D Navier-Stokes equation and the behavior of the direction of the velocity u/|u|. It is shown that the control of div(u/|u|) in a suitable L ^t/p (L ^x/q ) norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines. This work was supported in part by NSF Grant DMS-0607953.     

Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems

The interest in the use of quasimodes, or almost frequencies and almost eigenfunctions, to describe asymptotics for low‐frequency and high‐frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter ε, has been recently highlighted in many papers. In this paper we deal with the low frequencies for a Steklov‐type eigenvalue homogenization problem: we consider harmonic functions in a bounded domain of ?², and strongly alternating boundary conditions of the Dirichlet and Steklov type on a part of the boundary. The spectral parameter appears in the boundary condition on small segments T^ε of size O(ε) periodically distributed along the boundary; ε also measures the periodicity of the structure. We consider associated second‐order evolution problems on spaces of traces that depend on ε, and we provide estimates for the time t in which standing waves, constructed from quasimodes, approach their solutions u^ε(t) as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.