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Improved Adams-type inequalities and their extremals in dimension $2 m$

Albert-Ludwigs-Universität Freiburg, Mathematisches Institut. Abteilung für Reine Mathematik, Ernst-Zermelo-Str. 1, D-79104 Freiburg im Breisgau, Germany

E-mail Address: azahara.de.la.torre@math.uni-freiburg.de

Corresponding author.

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and

Gabriele Mancini

Sapienza Università di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Supported by INDAM - Istituto Nazionale di Alta Matematica, Via Antonio Scarpa 16, 00161 Roma, Italy

E-mail Address: gabriele.mancini@uniroma1.it

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https://doi.org/10.1142/S0219199720500431Cited by:9 (Source: Crossref)

Abstract

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space $H_{0}^{m} (Ω)$ , where $Ω$ is any bounded, smooth, open subset of $ℝ^{2 m}$ , $m \geq 1$ . Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

Keywords:

AMSC: 35J30, 35J91, 35B44, 35B33

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