The best constant of Sobolev inequality corresponding to the free boundary value problem for (-1)^M(d/dx)^<2M>(Theory,Applied Integrable Systems,<Special Issue>Joint Symposium of JSIAM Activity Groups 2007)

Abstract

M=1,2,3,…. For any function u(x) which satisfy u(x), u^<(M)>(x)∈L^2(-1,1), ∫^1_<-1>u(x)x^idx=0(0≤i≤M-1), there exists a positive constant C independent of u(x) such that the Sobolev inequality [numerical formula] hold. If M≤5, the best constant C(M) among such C is given as follows. C(M)=2<2M-1>(2(M-1))!(2M)!/(((M-1)!)^2(4M-1)!)(M≤5) Although the above equality is expected to hold for M≥6, its proof remains be unsolved.