A variational principle for the laplace's operator with application in the torsion of composite rods

The Ritz method with relaxed coordinate functions is used here to establish a minimizing sequence for an extended functional. The integrand of the functional involves discontinuous functions, which implies that the gradient of the related extremal curve or surface also admits discontinuities. It is discussed a systematic way to form an extended functional allowing the minimizing sequence to approach the exact solution displaying the discontinuities in the gradient of the extremal surface. The extra terms added to the classical functional are shown to be related to the Erdman-Weierstrass corner conditions. Finally the method is applied to the torsion of a composite rod, and the results compared with a solution obtained through the finite elements method.