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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