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The best constant of Sobolev inequality corresponding to clamped-free boundary value problem for \((-1)^M(d/dx)^{2M}\). (English) Zbl 1206.34048

Let \(H(M)\) be the Sobolev space \[ H(M):= \{u: u^{(M)}\in L^2(-1, 1), u^{(i)}(-1)= 0\text{ for }0\leq i\leq M-1\}, \] where \(M= 1,2,\dots\). The author determines the Green’s function \(G(x,y)\) to the boundary value problem \[ (-1)^M u^{(2M)}= f(x)\quad\text{for }-1< x< 1, \]
\[ u^{(i)}(-1)= 0,\quad u^{(M+ i)}(1)= 0\quad\text{for }0\leq i\leq M-1 \] and shows that \(G\) is the reproducing kernel for the Sobolev space \(H\) and that the best constant \(C_0\) in the Sobolev inequality
\[ \Biggl(\sup_{|y|\leq 1}|u(y)|\Biggr)^2\leq C\int^1_{-1} |u^{(M)}(x)|^2\,dx\quad\text{for }u\in H(M) \]
is given by \(C_0= G(1,1)\).

MSC:

34B27 Green’s functions for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
41A44 Best constants in approximation theory
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References:

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