The best constant of Sobolev inequality corresponding to clamped-free boundary value problem for (-1)M(d/dx)2M

The best constant of Sobolev inequality corresponding to clamped-free boundary value problem for (-1)^M(d/dx)^2M

Kazuo Takemura

Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 85, Number 8 (2009), 112-117.

Abstract

Green function of the clamped-free boundary value problem for (-1)^M(d/dx)^2M on the interval (-1,1) is obtained. Its Green function is a reproducing kernel for a suitable set of Hilbert space and an inner product. By using the fact, the best constant of Sobolev inequality corresponding to this boundary value problem is obtained as a function of M. The best constant is the maximal value of the diagonal value G(y,y) of Green function G(x,y).

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Primary Subjects: 34B05, 34B27, 46E22

Keywords: Sobolev inequality; best constant; Green function; reproducing kernel; LU decomposition

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pja/1254491215
Digital Object Identifier: doi:10.3792/pjaa.85.112
Zentralblatt MATH identifier: 05651157
Mathematical Reviews number (MathSciNet): MR2561901

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