## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 33, Number 2 (2009), 253-280.

### The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval

Yoshinori Kametaka, Atsushi Nagai, Kazuo Takemura, Kohtaro Watanabe, and Hiroyuki Yamagishi

#### Abstract

Green function of 2-point simple-type self-adjoint boundary value problem for 4-th order linear ordinary differential equation, which represents bending of a beam with the boundary condition as clamped, Dirichlet, Neumann and free. The construction of Green function needs the symmetric orthogonalization method in some cases. Green function is the reproducing kernel for suitable set of Hilbert space and inner product. As an application, the best constants of the corresponding Sobolev inequalities are expressed as the maximum of the diagonal values of Green function.

#### Article information

**Source**

Tsukuba J. Math., Volume 33, Number 2 (2009), 253-280.

**Dates**

First available in Project Euclid: 26 February 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1267209420

**Digital Object Identifier**

doi:10.21099/tkbjm/1267209420

**Mathematical Reviews number (MathSciNet)**

MR2605855

**Zentralblatt MATH identifier**

1209.34016

**Subjects**

Primary: 34B27: Green functions

Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 41A44: Best constants

**Keywords**

Green function Sobolev inequality best constant reproducing kernel symmetric orthogonalization method

#### Citation

Takemura, Kazuo; Yamagishi, Hiroyuki; Kametaka, Yoshinori; Watanabe, Kohtaro; Nagai, Atsushi. The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval. Tsukuba J. Math. 33 (2009), no. 2, 253--280. doi:10.21099/tkbjm/1267209420. https://projecteuclid.org/euclid.tkbjm/1267209420