An Improved Leray-Trudinger Inequality

2016/01/13 by Mallick, Arka, Tintarev, Cyril · 1 citation
#26D15 #35J92 #46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1601.03194

Abstract

In this article, we have derived the following Leray-Trudinger type inequality on a bounded domain Ω in ℝⁿ containing the origin. sup_u∈ W^1,n₀(Ω), I_n[u,Ω,R]≤ 1∫_Ω e^c_n(\frac|u(x)|E₂^β((|x|)/(R)))^(n)/(n-1) dx lt; +∞ , for some c_ngt;0 depending only on n. Here β= (2)/(n), I_n[u,Ω,R] := ∫_Ω|∇ u |ⁿdx- ((n-1)/(n))ⁿ∫_Ω\frac|u|ⁿ|x|ⁿE₁ⁿ((|x|)/(R))dx , R ≥ sup_{x∈ Ω}|x| and E₁(t) := log((e)/(t)), E₂(t) := log(eE₁(t)) for t∈ (0,1]. This improves an earlier result by Psaradakis and Spector. Also we have proved that, for any c>0 the above inequality is false, if we take β< (1)/(n).

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