Logarithmic Sobolev inequalities and the growth of norms

We show that many of the recent results on exponential integrability of Lip 1 functions, when a logarithmic Sobolev inequality holds, follow from more fundamental estimates of the growth of norms under the same hypotheses.

     

Global well-posedness of the Benjamin-Ono equation in low-regularity spaces

We prove that the Benjamin-Ono initial-value problem is globally well-posed in the Banach spaces , , of real-valued Sobolev functions.

     

Error estimates on anisotropic elements for functions in weighted Sobolev spaces

In this paper we prove error estimates for a piecewise average interpolation on anisotropic rectangular elements, i.e., rectangles with sides of different orders, in two and three dimensions.

Our error estimates are valid under the condition that neighboring elements have comparable size. This is a very mild assumption that includes more general meshes than those allowed in previous papers. In particular, strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses.

Moreover, we generalize the error estimates allowing on the right-hand side some weighted Sobolev norms. This extension is of interest in singularly perturbed problems.

Finally, we consider the approximation of functions vanishing on the boundary by finite element functions with the same property, a point that was not considered in previous papers on average interpolations for anisotropic elements.

As an application we consider the approximation of a singularly perturbed reaction-diffusion equation and show that, as a consequence of our results, almost optimal order error estimates in the energy norm, valid uniformly in the perturbation parameter, can be obtained.

     

Sharp Morrey-Sobolev inequalities and the distance from extremals

Quantitative versions of sharp estimates for the supremum of Sobolev functions in , , with remainder terms depending on the distance from the families of extremals, are established.

     

Regularity estimates for elliptic boundary value problems in Besov spaces

We consider the Dirichlet problem for Poisson's equation on a nonconvex plane polygonal domain . New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new interpolation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary value problems. Some new sharp finite element error estimates are deduced.

     

Inverse-type estimates on -finite element spaces and applications

This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to -finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The assumptions on the -finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of -boundary element method discretisations of integral equations, with element-wise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.

     

On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality

where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality,

where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete:

and the corresponding eigenfunction spaces are

     

Finite time blow up for a Navier-Stokes like equation

We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so-called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space . We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.

     

Self Improving Sobolev-Poincaré Inequalities,Truncation and Symmetrization

In Martín et al. (J Funct Anal 252:677–695, 2007) we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in . In this paper we extend our method to Sobolev functions that do not vanish at the boundary. This paper is in final form and no version of it will be submitted for publication elsewhere.     

On characterizations of commutativity of C-algebras

We give three kinds of characterizations of the commutativity of C- algebras. The first is the one from operator monotone property of functions regarded as the nonlinear version of Stinespring theorem, the second one is the characterization of commutativity of local type from expansion formulae of related functions and the third one is of global type from multiple positivity of those nonlinear positive maps induced from functions.

     

Sharp Sobolev inequalities in the presence of a twist

Let be a smooth compact Riemannian manifold of dimension . Let also be a smooth symmetrical positive -tensor field in . By the Sobolev embedding theorem, we can write that there exist such that for any ,

where is the standard Sobolev space of functions in with one derivative in . We investigate in this paper the value of the sharp in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.

     

Remark on ``A problem of prescribing Gaussian curvature on " [Proc. Amer. Math. Soc. 129 (2001), no. 12, 3757-3758]

In this note, we remark on a 2001 paper of S. Goyal and V. Goyal. The main result of this work is that they used some elementary method to find a class of functions for which the solutions to

on can be obtained. We observe that this class of functions that they studied is actually the trivial one, i.e. the class of positive constant functions.

     

Isometric weighted composition operators on weighted Banach spaces of type

We characterize those weighted composition operators on weighted Banach spaces of holomorphic functions of type which are an isometry.

     

Blow-up examples for second order elliptic PDEs of critical Sobolev growth

Let be a smooth compact Riemannian manifold of dimension , and be the Laplace-Beltrami operator. Let also be the critical Sobolev exponent for the embedding of the Sobolev space into Lebesgue's spaces, and be a smooth function on . Elliptic equations of critical Sobolev growth such as

have been the target of investigation for decades. A very nice -theory for the asymptotic behaviour of solutions of such equations has been available since the 1980's. The -theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of . It was used as a key point by Druet to prove compactness results for equations such as . An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of . We present such examples in this article.

     

The combinatorics of Bernstein functions

A construction of Bernstein associates to each cocharacter of a split -adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz' conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of ; one example can be used to verify Kottwitz' conjecture for a special class of Shimura varieties (the ``Drinfeld case'). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the ``-admissible' set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.

     

Continuity of the maximal operator in Sobolev spaces

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces , . As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

     

A remark on Calderón-Zygmund classes and Sobolev spaces

We show how the Sobolev space may be characterized in terms of the local behavior of its members. We use the local -classes introduced by Calderón and Zygmund.

     

Hyperbolic conservation laws with stiff reaction terms of monostable type

In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type

is studied. Solutions of Cauchy problems of the above equation with initial value are proved to converge, as , to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed . The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving 0$"> is found to originate from the behavior of traveling waves of the above system with viscosity regularization.

     

An embedding theorem of Sobolev type for an operator with singularity

We discuss spaces of Sobolev type which are defined by the operator with singularity: , where and . This operator appears in a one-dimensional harmonic oscillator governed by Wigner's commutation relations. We study smoothness of and continuity of () where is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.

     

Hardy inequalities with optimal constants and remainder terms

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces and in higher-order Sobolev spaces on a bounded domain can be refined by adding remainder terms which involve norms. In the higher-order case further norms with lower-order singular weights arise. The case being more involved requires a different technique and is developed only in the space .