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

2016/07/26 by Parini, Enea, Ruf, Bernhard · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1607.07681

Abstract

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality sup \∫_Ωexp(α |u|^(N)/(N-s)) | u ∈ \widetildeW^s,p₀(Ω), [u]_W^s,p(ℝ^N)≤ 1 \lt; + ∞. Here Ω is a bounded domain of ℝ^N (N≥ 2), s ∈ (0,1), sp = N, \widetildeW^s,p₀(Ω) is a Sobolev-Slobodeckij space, and [⋅]_W^s,p(ℝ^N) 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,+∞).

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