Numerical computation of least constants for the Sobolev inequality (Q580651)

This paper considers the Sobolev space \(H^ m(G)\) for a domain G in \(R^ n\) which is the completion of \(C^ m(G)\) (the space of m-times continuously differentiable functions on G) with respect to the Sobolev norm. This norm is defined to be \[ \| f\|^ 2_{m,G}=\sum_{| p| \leq m}\| D^ pf\|^ 2 \] where \(\| \|\) denotes the norm of \(L_ 2(G)\). The paper finds least constants \(C_ m(G)\) for various choices of G where \[ \| f\|_{\infty}\leq C_ m(G)\| f\|_{m,G} \] and \(\|.\|_{\infty}\) is the supremum norm on G. The various domains G considered include intervals of the real numbers and use is made of reproducing kernels which are functions K from \(\bar G\) (the closure of G) to \(H^ m(G)\).