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- W2962829538 abstract "Abstract In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into L ∞ ( R N ) and those that embed into slightly larger target spaces. Concerning the former, we show that for k ∈ { 1 , … , N − 1 } , N − k even, one has an optimal constant c k > 0 such that ‖ u ‖ L ∞ ≤ c k ∫ | ∇ k ( − Δ ) ( N − k ) ∕ 2 u | for all u ∈ C c ∞ ( R N ) (the case k = N was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams’ higher order inequality of J. Moser: For Ω ⊂ R N , m ∈ N and p = N m , there exists A > 0 and optimal constant β 0 > 0 such that ∫ Ω exp ( β 0 | u | p ′ ) ≤ A | Ω | for all u such that ‖ ∇ m u ‖ L p ( Ω ) ≤ 1 , where ‖ ∇ m u ‖ L p ( Ω ) is the traditional semi-norm on the space W m , p ( Ω ) ." @default.
- W2962829538 created "2019-07-30" @default.
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- W2962829538 date "2018-12-01" @default.
- W2962829538 modified "2023-09-29" @default.
- W2962829538 title "Best constants for two families of higher order critical Sobolev embeddings" @default.
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- W2962829538 doi "https://doi.org/10.1016/j.na.2018.04.027" @default.
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