In this paper, we investigate the existence and characterizations of the Fréchet derivatives of the solution to time-harmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique the factorization of the difference of the far-field pattern for two different scatterers introduced by Kress and Païvarinta to establish Fréchet differentiabil...

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...

"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1,61, 389) with so much ingenuity...and other equations occurring in the literature of mathematics, do not di...