Abstract
The Green function of the free boundary value problem for $(-1)^{M}(d/dx)^{2M}$ is found by using Whipple’s formula. The Green function is constructed through so-called symmetric orthogonalization method under a suitable solvability condition. Its Green function is a reproducing kernel for a suitable set of Hilbert space and an inner product. By using the fact, we compute the best constant ( $M=1,2,3,4,5$ ) and a family of the best functions for a Sobolev inequality. It is possible for us to expect the best constant of the Sobolev inequality, but the proof has not been completed for $M\geq6$ in the present paper.
Highlights
1 Introduction For M =, . . . , we introduce the Sobolev space
The purpose of the present paper is to find a supremum of the Sobolev functional given by
In Section, we show that the Green function is the reproducing kernel for H and (·, ·)M
Summary
Kazuo Takemura1*, Atsushi Nagai[2], Yoshinori Kametaka[3], Kohtaro Watanabe[4] and Hiroyuki Yamagishi[5]
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