Rocky Mountain Journal of Mathematics

On weak continuity of the Moser functional in Lorentz-Sobolev spaces

Robert Černý

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Abstract

Let $B(R)\subset \mathbb{R}^n $, $n\in \mathbb{N} $, $n\geq 2$, be an open ball. By a result from~\cite {AdT}, the Moser functional with the borderline exponent from the Moser-Trudinger inequality fails to be sequentially weakly continuous on the set of radial functions from the unit ball in $W_0^{1,n}(B(R))$, only in the exceptional case of sequences acting like a~concentrating Moser sequence.

We extend this result into the Lorentz-Sobolev space $W_0^1L^{n,q}(B(R))$, with $q\in (1,n]$, equipped with the norm $$ ||\nabla u||_{n,q}:= ||t^{1/n-1/q}|\nabla u|^*(t)||_{L^q((0,|B(R)|))}. $$ We also consider the case of a nontrivial weak limit and the corresponding Moser functional with the borderline exponent from the concentration-compactness alternative.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 3 (2017), 757-788.

Dates
First available in Project Euclid: 24 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1498269810

Digital Object Identifier
doi:10.1216/RMJ-2017-47-3-757

Mathematical Reviews number (MathSciNet)
MR3682148

Zentralblatt MATH identifier
1381.26020

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
Sobolev spaces Lorentz-Sobolev spaces Moser-Trudinger in­equal­ity concentration-compactness principle sharp constants

Citation

Černý, Robert. On weak continuity of the Moser functional in Lorentz-Sobolev spaces. Rocky Mountain J. Math. 47 (2017), no. 3, 757--788. doi:10.1216/RMJ-2017-47-3-757. https://projecteuclid.org/euclid.rmjm/1498269810


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