On the dual boundary conditions and the nonnegative square root of the second derivative operator (Q1566842)

The study of elastic systems with structural damping gives rise to differential equations of the form \[ {\partial^2u\over\partial t^2}+ Au+ rA^{1/2}{\partial u\over\partial t}= 0, \] where \(A\) is a nonnegative selfadjoint differential operator and \(A^{1/2}\) is the square root of \(A\). The author describes this square root for the case that \(A\) is the second derivative in \([0,2\pi]\) with boundary conditions such that it becomes a selfadjoint nonnegative operator in \(L_2(0, 2\pi)\). The square root is described by means of a certain singular integral operator the kernel of which may have singularities at the boundary and on the diagonal of \([0,2\pi]\times [0,2\pi]\). It turns out that \(A^{1/2}u\) is in some sense a harmonic conjugate of \(u'\), and it is calculated explicitly for different kinds of boundary conditions.