On Weighted Poincaré Inequalities

We are concerned with the problem of finding sharp summability conditions on the weights which render certain weighted inequalities of Poincaré - type true. The conditions we find consist of proper integral balances between the growths of the rearrangements of the weights.     

Weighted Hardy inequalities

For bounded Lipschitz domains D in it is known that if 1<p<∞, then for all β[0,β₀), where β₀=p−1>0, there is a constant c<∞ with

for all . We show that if D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β₀ now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for W^1,p(Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality

c=c(n,p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains.     

On the instability of taylor and dean flows under the influence of a circular magnetic field

Résumé On discute l'influence d'un champ magnétique circulaire sur la stabilité d'un fluide visqueux et conducteur de l'électricité en mouvement entre deux cylindres circulaires coaxiaux. Supposant que l'espacement entre les cylindres est petit par rapport à leurs rayons, on donne des résultats sur deux sortes de mouvement du fluide basique, produites par la rotation des cylindres et par un gradient de pression autour des cylindres. On démontre que le champ magnétique peut avancer ou retarder l'accrochage de l'instabilité, suivant les circonstances.     

Norms of embeddings of logarithmic Bessel potential spaces

Let be a subset of with finite volume, let and let be a Young function with for large . We show that the norm on the Orlicz space is equivalent to

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in which are sharp in their dependences on provided that is large enough.

     

CsF–Si(OEt)4 catalyzed addition of (hetero)aromatic amides to ethyl acrylate

The conjugate addition of some (hetero)aromatic amides to an α,β‐unsaturated ester (ethyl acrylate) proceeds efficiently in the presence of an equimolar amount of the CsF–Si(OEt)₄ system to afford the corresponding ethyl esters of N‐substituted β‐amino acids. Copyright © 2001 John Wiley & Sons, Ltd.     

Sobolev inequalities of exponential type

We give sufficient conditions for domains to satisfy Sobolev inequalities of single exponential type. Earlier work in this area imposed more stringent conditions on the domains and is thus contained in our results. Moreover, the class of functions considered is based onL ⁿ log ^an L witha<1−1/n, n being the dimension of the underlying space. The limiting casea=1−1/n gives rise to an inequality of double exponential type which is shown to be valid in a large class of irregular domains. This inequality is new even in smooth domains. The second author was partially supported by a grant from Magnus Ehrnrooth Foundation.     

Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.     

Eigenvalue asymptotics and a non-linear Schrödinger equation

The equation 1 $$ - \Delta u + qu + f(x,u) = \lambda u, u \in W^{1,2} (\mathbb{R}^N )$$ is considered, whereq is bounded below andq(x)→∞ as |x|→∞. Under appropriate conditions on the perturbation termf(x, u) it is shown that given anyr>0, (*) has an infinite sequence (λ _{n, r} ) _{n ∈ N} of eigenvalues, eachλ _{n, r} being associated with an eigenfunctionu _n,r which satisfies \(\smallint _{R^N } \left| {u_{n,r} } \right|^2 = r^2 \) . Information about the behaviour ofλ _{n, r} for largen is provided. The proofs rely on the compactness of the embedding of a certain weighted Sobolov space in anL ^p space; this is proved in §2.