_{1}

The best constant of discrete Sobolev inequality on the truncated tetrahedron with a weight which describes 2 kinds of spring constants or bond distances. Main results coincides with the ones of known results by Kametaka et al. under the assumption of uniformity of the spring constants. Since the buckyball fullerene C60 has 2 kinds of edges, destruction of uniformity makes us proceed the application to the chemistry of fullerenes.

Sobolev inequality known as Sobolev embedding theorem plays an important role in the theory of PDEs. Brezis [1, Chap.IX] gave some constant of Sobolev inequality, and mentioned that the best constant was known and complex. Talenti [

Kametaka and his coworkers studied the best constant of Sobolev inequality in view of the boundary value problem [

On the other hand, in chemistry of fullerenes [

This article concerns with the best constant of discrete Sobolev inequality on T4 with 2 kinds of spring constants, in other words, a weighted T4 graph. The results of Kametaka school for R4 [

Consider the truncated tetrahedron T4. It has 12 vertices, and let us number the vertices 0, 1, …, 11 as in

Define the bond matrix

Note that

Here,

By use of the weighted Laplacian defined as above, the Sobolev energies are written as follows:

The eigenvalues of

where

For the Green matrix, there exists a unique matrix

Theorem 1. There exists a positive constant

holds. Among such

Theorem 2. There exists a positive constant

holds. Among such

Remark.

R4 | R6 | R8 | R12 | R20 | |
---|---|---|---|---|---|

The best constant | 3/16 ? 0.1875 | 29/96 ? 0.30208 | 13/72 ? 0.18056 | 137/300 ? 0.45667 | 7/36 ? 0.19444 |

T4 | T6 | T8 | T12 | T20 | |
---|---|---|---|---|---|

The best constant | 301/720 ? 0.41806 | 173/288 ? 0.60069 | 1019/2016 ? 0.50546 | - | 239741/376200 ?0.63727 |

Let

Note that

Definition. For any

Lemma. For every

Remark. So,

Proof of Lemma. Since

Proof of Theorems. Applying the Schwarz inequality to the reproducing equality, we have

Using

Then we obtain discrete Sobolev inequality:

Then, for

Combining it with the trivial inequality

We obtain the conclusion of Theorem 1. Theorem 2 is similarly proved.

Kametaka school says that the high symmetry of Rn or Tn allows us to compute the exact expression of the best constant. However, the introduction of our weight does not destroy the computability of this problem because our weighted Laplacian is still symmetric matrix. Whether our model with weight is appropriate or not is another problem. It depends on what kind of problem we want to apply our model to.

And, after this article, the author wish to study the Tn for n = 6, 8, 12, 20, and application to the interaction of fullerene and another molecules. The high symmetry move us to its beauty however, the destruction of the symmetry also fascinates us.

The author thanks Prof. T. Masuda for his suggestion to read one of the papers of Kametaka school on the best constant of discrete Sobolev inequality, and also thanks his friends S. Fuchigami, R. Inoue and S. Minami for helpful discussion.

Yoshikatsu Sasaki, (2015) The Best Constant of Discrete Sobolev Inequality on a Weighted Truncated Tetrahedron. World Journal of Engineering and Technology,03,149-154. doi: 10.4236/wjet.2015.33C022