A Cacciopoli-type inequality to prove coercivity of a bilinear form associated with spatial hysteresis internal damping for an Euler–Bernoulli beam

We prove an inequality that resembles Cacciopoli inequalities in that it bounds the norm of the derivative of a function by using the norm of the function. Unlike in Cacciopoli inequalities, there is no restriction on the function, a fact made up for by adding an extra term to the norm of the function. The inequality arose in the proof that a bilinear form associated with spatial hysteresis internal damping for an Euler–Bernoulli beam is coercive.