Partial Schauder estimates for second-order elliptic systems

Abstract

Second order elliptic equations and systems are among the most important types of partial differential equation(PDEs). The classical Schauder’s theory for this type of equations has played an essential role in the study of linear and non-linear elliptic equations, which reveals that if all the coefficients and data are Holder continuous in all variables, then the solutions and all their derivatives up to second order will also be Holder continuous in all the variables. The main objective of this research is to obtain a class of pointwise estimates, called partial Schauder estimates, for second-order elliptic systems. The desired result will show that if the inhomogeneous term f ^a is Holder continuous in the x_n direction, then the u_x , ...,u_x_n derivatives are also Holder continuous.