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.
Journal of Mathematical Analysis and Applications
Schröder, B. S.,
Walters, J. B.,
Evans, K. A.
(2016). A Cacciopoli-Type Inequality to Prove Coercivity of a Bilinear Form Associated with Spatial Hysteresis Internal Damping for an Euler-Bernoulli Beam. Journal of Mathematical Analysis and Applications, 425(1), 520-535.
Available at: https://aquila.usm.edu/fac_pubs/15691