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1.
Let m and n be positive integers with n2 and 1mn−1. We study rearrangement-invariant quasinorms R and D on functions f: (0, 1)→
such that to each bounded domain Ω in
n, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality R(u*(|Ω| t))CD(|mu|* (|Ω| t)), uCm0(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which D need not be rearrangement-invariant, R(u*(|Ω| t))CD((d/dt) ∫{x
n : |u(x)|>u*(|Ω| t)} |(u)(x)| dx), uC10(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that R cannot be replaced by an essentially larger quasinorm and D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger. 相似文献
2.
Let H, V be two real Hilbert spaces such that VH with continuous and dense imbedding, and let FC1(V) be convex. By using differential inequalities, a close-to-optimal ultimate bound of the energy is obtained for solutions in to u″+cu′+bu+F(u)=f(t) whenever . 相似文献
3.
Donglong Li Zhengde Dai Xuhong Liu 《Journal of Mathematical Analysis and Applications》2007,330(2):934-948
In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)
ut=ρu−Δφ(u)−(1+iγ)Δu−νΔ2u−(1+iμ)|u|2σu+αλ1(|u|2u)+β(λ2)|u|2