A note on the Lp-Sobolev inequality

2023/12/31 by Deng, Shengbing, Tian, Xingliang
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.00464

Abstract

The usual Sobolev inequality in ℝ^N, asserts that ‖∇ u‖_L^p(ℝ^N) ≥ S‖u‖_{L^{p^*}(ℝ^N)} for 10 independent of Ω such that ‖∇ u‖^p_L^p(Ω) -S^p‖u‖^p_L^{p^*}(Ω) ≥ C|Ω|^-\fracγp^*(p-1) ‖u‖_L^p_w(Ω)^γ‖ u‖_L^{p^*}(Ω)^p-γ, for all u∈ C^∞₀(Ω)∖\0\, where γ=max\2,p\, p=p^*(p-1)/p, and ‖⋅‖_L^p_w(Ω) denotes the weak L^p-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in ℝ^N.

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