Explicit Form Of Extremal Functions In The Embedding Constant Problem For Sobolev SpacesI

TL;DR

This paper derives explicit extremal functions and formulas for embedding constants in Sobolev spaces, revealing their relation to Legendre polynomials and spectral problems, with detailed results for specific cases k=3 and k=5.

Contribution

It provides explicit extremal functions and formulas for Sobolev embedding constants, connecting them to Legendre polynomials and spectral problems, advancing understanding of these constants.

Findings

Explicit extremal functions for embedding constants are derived.

Formulas for specific cases k=3 and k=5 are obtained.

A connection between embedding constants and spectral problems is established.

Abstract

The embedding constants of the Sobolev spaces $\mathring{W}^n_2[0;1] \hookrightarrow \mathring{W}^k_\infty[0; 1]$ ($0\leqslant k \leqslant n-1$) are studied. A relation of the embedding constants with the norms of the functionals $f\mapsto f^{(k)}(a)$ in the space $\mathring{W}^n_2[0;1]$ is given. An explicit form of the functions $g_{n;k}\in \mathring{W}^n_2[0;1]$ on which these functionals attain their norm is found. These functions are also to be extremal for the embedding constants. A relation of the embedding constants to the Legendre polynomials is put forward. A detailed study is made of the embedding constants with k = 3 and k = 5: we found explicit formulas for extreme points, calculate global maximum points, and give the values of the sharp embedding constants. A link between the embedding constants and some class of spectral problems with distribution coefficients is discovered.