The best constant of Sobolev inequality corresponding to anti-periodic boundary value problem

Abstract

In this paper we establish the best constant of $\mathcal{L}^{p}$ Sobolev inequality for a function with anti-periodic boundary conditions. The best constant is expressed by $\mathcal{L}^q$ norm of $(M-1)$-th order Euler polynomial. Lyapunov-type inequality for certain higher order differential equation including 1-dim $p$-Laplacian is obtained by the usage of this constant.

Keywords

• $l^{p}$ sobolev inequality
• best constant
• green function
• euler polynomial
• lyapunov-type inequality.