Distributional solution concepts for the Euler-Bernoulli beam equation with discontinuous coefficients

Abstract

We study existence and uniqueness of distributional solutions to the differential equation of the Euler-Bernoulli rod with discontinuous coefficients and right-hand side. Upon checking the validity of a solution the occurring products of singular coefficients with the distributional solution have no obvious meaning. When interpreted on the most general level of the so-called hierarchy of distributional products, it turns out that existence of a solution forces a minimum regularity to hold. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of having a discontinuous displacement function as a solution. We also give conditions for unique solvability.