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The best constant of Sobolev inequality in an \(n\)-dimensional Euclidean space. (English) Zbl 1100.46021

The best constant of the Sobolev inequality in an \(n\)-dimensional Euclidean space is found by means of the theory of reproducing kernels and Green functions. The concrete form of the best constant is also found in the case of the Sobolev space \(W^2(\mathbb{R}^n)\) \((n=2,3)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
26D15 Inequalities for sums, series and integrals
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References:

[1] H. Brezis, Analyse fonctionnelle , Masson, Paris, 1983.
[2] Y. Kametaka, K. Watanabe, A. Nagai and S. Pyatkov, The best constant of sobolev inequality in an \(n\) dimensional eucledean space, Sci. Math. Jpn. 61 (2005), no.1, 15-23. · Zbl 1079.46023
[3] C. Morosi and L. Pizzocchero, On the constants for some Sobolev imbeddings, J. Inequal. Appl. 6 (2001), no. 6, 665-679. · Zbl 1089.46509 · doi:10.1155/S1025583401000418
[4] K. Watanabe, T. Yamada and W. Takahashi, Reproducing kernels of \(H^ m(a,b)\) \((m=1,2,3)\) and least constants in Sobolev’s inequalities, Appl. Anal. 82 (2003), no. 8, 809-820. · Zbl 1051.46021 · doi:10.1080/0003681031000152541
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