Abstract
This paper clarifies the hierarchical structure of the sharp constants for the discrete Sobolev inequality on a weighted complete graph. To this end, we introduce a generalized-graph Laplacian A = I − B on the graph, and investigate two types of discrete Sobolev inequalities. The sharp constants C 0 ( N ; a ) and C 0 ( N ) were calculated through the Green matrix G ( a ) = ( A + a I ) − 1 ( 0 < a < ∞ ) and the pseudo-Green matrix G ∗ = A † . The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of A. Based on this new discovery, we provide the first proof that each set of the sharp constants { C 0 ( n ; a ) } n = 2 N and { C 0 ( n ) } n = 2 N satisfies a certain hierarchical structure.
Highlights
The sharp constant and a family of best functions for the Sobolev inequality, kuk Lq (Rn ) ≤ C k∇uk L p (Rn ) ∀u ∈ W 1,p (Rn ) = {u ∈ Lq (Rn ), ∇u ∈ L p (Rn )}, (1)were independently discovered by Aubin [1] and Talenti [2] in the case 1 < p < n, q = np/(n − p).They computed the sharp constant using a symmetric rearrangement and found the family of best functions that satisfy the equality in (1)
We mainly investigate the case p ≤ 2, q = ∞ and compute the sharp constants for the Sobolev inequalities using the Green functions corresponding to various boundary value problems
The purpose of this paper is to extend the results on a complete graph by Yamagishi et al [8] to a weighted graph
Summary
The sharp (smallest) constant and a family of best functions for the Sobolev inequality, kuk Lq (Rn ) ≤ C k∇uk L p (Rn ). Were independently discovered by Aubin [1] and Talenti [2] in the case 1 < p < n, q = np/(n − p) They computed the sharp constant using a symmetric rearrangement and found the family of best functions that satisfy the equality in (1). We mainly investigate the case p ≤ 2, q = ∞ and compute the sharp constants for the Sobolev inequalities using the Green functions corresponding to various boundary value problems. This is a unique approach that remarkably differs from symmetric rearrangement.
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