This chapter deals with linear equations while the remainder is devoted to nonlinear equations. The part dealing with linear equations commences with some simple, novel estimates for the harmonic equation and then discusses some estimates for the equations of linear elasticity. The elasticity estimates are broadly of two types: the first class deals with cylinders/strips where – usually but not exclusively – zero displacement is prescribed (Dirichlet conditions) on the lateral boundary/ies, while the second deals with such regions where zero traction (Neumann) conditions are prescribed on the lateral boundary. The chapter obtains qualitative estimates for cylindrical and other regions whose lateral boundaries are subject to displacement boundary conditions. The cylinder consists of homogeneous, isotropic, linear elastic material and its lateral boundary is displacement-free.