On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces

By Enea Parini, Bernhard Ruf. Journal d'Analyse Mathematique, 2016

62 citations recorded by the source corpus.

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality sup{∫Ωexp(α∣u∣NN−s)∣u∈W˜0s,p(Ω),[u]Ws,p(ℝN)≤1}<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{sup}}\left\{ {\left. {\int_{\rm{\Omega }} {{\rm{exp}}\left( {\alpha \;{\rm{|}}u{{\rm{|}}^{{N \over {N - s}}}}} \right)} } \right|u \in \widetilde{W}_0^{s,p}({\rm{\Omega }}),\;{{[u]}_{{W^{s,p}}({\mathbb{R}^N})}} \le 1} \right\} < \infty $$\end{document}. Here Ω is a bounded domain of ℝN (N ≥ 2), s ∊ (0, 1), sp = N, W˜0s,p(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{W}_0^{s,p}({\rm{\Omega }})$$\end{document} is a Sobolev-Slobodeckij space, and [⋅]Ws,p(ℝN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[ \cdot ]_{{W^{s,p}}({\mathbb{R}^N})}}$$\end{document} is the associated Gagliardo seminorm. We exhibit an explicit exponent α*s,N > 0, which does not depend on Ω, such that the Moser-Trudinger inequality does not hold true for α ∊ (α*s,N, +∞).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c7af17b4e8096dc02a085eeecfbb41eb464d104c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Lenka Slav'ikov'a
  2. L. Pick
  3. Andrea Cianchi
  4. A. Alberico
  5. Caifeng Zhang
  6. Iwona Chlebicka
  7. Cristina Tarsi
  8. Joaquim Martín

In Mathematical Physics

  1. Lenka Slav'ikov'a
  2. L. Pick
  3. Andrea Cianchi
  4. A. Alberico
  5. Caifeng Zhang
  6. Iwona Chlebicka
  7. P. Hasto
  8. Joaquim Martín

Citations in the corpus