On inequalities of Bliss-Moser type with loss of compactness in ℝ^N

2025/01/06 by Guo, Yanyan, Luo, Huxiao, Ruf, Bernhard
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.03168

Abstract

We prove the following Limiting Bliss inequalities sup_{v(0) = 0, ∫0¹|v'|^Ndx=1} ∫₀¹ e^β(log(e)/(s))\fracv^N(s)s^N-1ds≤ C(N,β), \hbox for β≤ 1 The inequalities are optimal with respect to β≤ 1; there is compactness for β<1, and along the infinitesimal Moser sequence for β= 1. Moreover, we show that the improved inequalities sup_{v(0) = 0, ∫0¹|v'|^Ndx=1} ∫₀¹ e^(log(e)/(s)+γloglog(e)/(s))\fracv^N(s)s^N-1ds≤ C(N,γ) hold for γ≤1, and for γ=1 the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible. The second result extends the result in [J. M. do Ó, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from N=2 to general dimensions N≥2.

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