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Table 2 Computed values of \(\pmb{C_{p}(\Omega)}\) for \(\pmb{\Omega=(0,1)^{2}}\) and \(\pmb{q=2}\) . The numbers of division n are shown in the corresponding parentheses. Theorem 3.1 cannot be used for \(\pmb{p>4}\) when \(\pmb{N=2}\)

From: Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

p Theorem  3.1 Theorem  3.2 Theorem  3.3 [ 10 , Lemma 2.3] [ 19 , Corollary D.1]
3 2.553767 (16) 4.423506 (256) 2.647076 (16) 1.272533 1.291703 × 104
4 2.506629 (4) 5.656855 (256) 3.098954 (16) 1.553774 1.809271 × 104
5 - 5.721912 (64) 3.527578 (16) 1.841950 2.275458 × 104
6 - 5.802230 (64) 3.922709 (16) 2.135792 2.701890 × 104
7 - 6.245674 (64) 4.288114 (16) 2.434362 3.096661 × 104
8 - 6.727172 (64) 4.628497 (16) 2.736941 3.465528 × 104
9 - 7.127190 (64) 4.947849 (16) 3.042967 3.812726 × 104
10 - 7.464264 (64) 5.249352 (16) 3.351991 4.141471 × 104
20 - 9.162396 (16) 7.659208 (16) 6.549949 6.789009 × 104
30 - 10.202188 (64) 9.485455 (16) 9.856546 8.800592 × 104
40 - 11.632217 (64) 10.640059 (64) 13.218367 1.048141 × 105
50 - 12.907885 (64) 12.020066 (64) 16.613831 1.195208 × 105
60 - 14.069728 (64) 13.258962 (64) 20.031993 1.327453 × 105
70 - 15.143396 (64) 14.392550 (64) 23.466517 1.448540 × 105
80 - 16.146231 (64) 15.443710 (64) 26.913400 1.560849 × 105