Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations (Q426450)

The authors prove some Lyapunov-type inequalities for the equation NEWLINE\[NEWLINE u^{(2n)} (x) + r(x)u(x) = 0, \;a \leq x \leq b, NEWLINE\]NEWLINE where \(r(\cdot)\) is a continuous and nonnegative function in \([a,b]\) and for five types of boundary conditions: Dirichlet, periodic, anti-periodic, Neumann and Dirichlet-Neumann. The proof uses the best constant of some Sobolev-type inequalities for higher order derivatives where the Riemann zeta function plays an important role. More precisely, this best constant is associated with the study of some minimization problems for functionals of the form NEWLINE\[NEWLINE \frac{\int_{a}^{b} | u^{(n)} (x) | ^{2} \;dx} {\left ( \sup_{a \leq x \leq b} | u(x) | \right ) ^{2}} NEWLINE\]NEWLINE where the functions \(u(\cdot)\) belong to an appropriate Sobolev space with the corresponding boundary conditions.