The Best Constant of $L^p$ Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II

Abstract

Let $M = 2m$ $(m = 1,2,\ldots)$. In [1] the best constant of $L^p$ Sobolev inequality $$ \sup_{-1\leq x\leq 1}\vert u(x)\vert\leq C\Biggl( \int_{-1}^{1}\vert u^{(M)}(x)\vert^{p}dx\Biggr)^{1/p} $$ was obtained for $u$ satisfying $u, u^{(M)} \in L^{p}(-1,1)$ and $u^{(2i)}(\pm 1) = 0$ $(0\leq i\leq [(M-1)/2])$. On the other hand, for the case $M$ is odd, up to now, only the case $M =1$ was treated for technical difficulty; see [2]. This paper treats the case $M =3$ with different two approach, one is based on the property of the function associated with certain Green function and another is on the property of function space. For the latter approach, symmetrizations of functions play an important role.