THE BEST CONSTANT OF L^p SOBOLEV INEQUALITY CORRESPONDING TO DIRICHLET-NEUMANN BOUNDARY VALUE PROBLEM

We have obtained the best constant of the following L^p Sobolev inequality sup _0≤y≤1| u^(j)(y)| ≤C (∫ ₀¹ | u^(M)(x)| ^p dx)^1/p , where u is a function satisfying u^(M) ∈ L^p(0, 1), u⁽²ⁱ⁾(0) = 0 (0 ≤i ≤ [(M − 1)/2]) and u⁽²ⁱ⁺¹⁾(1) = 0 (0 ≤ i ≤ [(M − 2)/2]), where u⁽ⁱ⁾ is the abbreviation of (d/dx)ⁱu(x). In [9], the best constant of the above inequality was obtained for the case of p = 2 and j = 0. This paper extends the result of [9] under the conditions p > 1 and 0 ≤ j ≤ M −1. The best constant is expressed by Bernoulli polynomials.