Schauder Estimates For Solutions To Several Types Of Partial Differential Equations

Schauder estimates is one of the important results of partial differential equations.It provides an essential foundation for study of the existence and regularity of solutions of nonlinear partial differential equations.There have been many methods for the proof of Schauder estimates for elliptic and parabolic equations,and the proofs usually involve lengthy and complicated analysis.Xujia Wang gave a very elegant,brief,elementary proof of the Schauder estimates in 2006.In this paper,we study Xujia Wang's paper and examine Schauder estimates of several types of partial differential equations.Our discussions are based on the idea and methods of Xujia Wang and we provide all details of analysis in the proof.In our discussions,we use the properties of solutions of Laplace's equation and Heat equation,and the maximum principle for elliptic and parabolic equations.We give the proof of the estimates for the modulus of continuity of second order derivatives of solutions when the nonhomogeneous term on the right hand of the equations and coefficients are Dini continuous for three types of second order partial differential equations: elliptic equations with variable coefficients,parabolic equations with constant coefficients,parabolic equations with variable coefficients.We get results consistent with the classical Schauder interior estimates when the nonhomogeneous term on the right hand of the equations and coefficients are H?lder continuous.In Chapter 4,we use the results about the Schauder estimates to prove the regularity of solutions for some important partial differential equations.We specifically study three types of decomposable equations.After decomposition,the first one is an ordinary differential equation and an elliptic equation,the second one is a parabolic equation and an elliptic equation,the third one is an integral equation and a parabolic equation.For the second type of decomposable equation,We specifically consider the cases of the parabolic equation and the elliptic equation are Dirichlet's boundary conditions or oblique derivative boundary conditions.After appropriate transformation,the Schauder estimates are applied successively to the two equations,and finally the regularity of solutions and the corresponding estimates are obtained.