Parabolic Schauder Estimates Of 2m Order On Banded Manifolds

Schauder estimates play a crucial role in the theoretical study of partial differential equations and is the basis for studying the existence,uniqueness and regularity of solutions.So far,many scholars have studied elliptic and parabolic differential equations,and draw many classic conclusions.In this article,I carefully studied the second chapter of Lamm’s paper published in 2001,and modified the description and proof process of some lemmas in his article,such as lemma 2.3.6 and lemma 2.3.19.On this basis,I used my own ideas to summarize and organize the schauder estimates of 2m order parabolic differential equations with a strong elliptic main part under different boundary value conditionsThis article is mainly divided into five chapters to introduce.The first chapter is the preliminaries,which mainly introduces some basic theories and common concepts needed to study schauder estimates.The second chapter focuses on the introduction of Garding’s inequality,and uses it to study the L2 interior estimates,boundary estimates and global estimates of 2m order elliptic differential equations.The third chapter introduces the function spaces involving time,in this kind of space,we mainly uses the energy estimates method to study the L2 regularity and higher regularity of the 2m order parabolic differential equation.The fourth chapter is the core content of this paper,firstly,with the help of interpolation estimation,absorption lemma,etc.,we prove the H(?)lder interior estimates and global estimates of 2m order linear parabolic differential equations with constant coefficients and Cauchy and Dirichlet problems.On this basis,using the solidification coefficient method,we prove the schauder estimates of 2m order linear parabolic differential equations with general coefficients under various boundary value conditions.Finally,we transfer the obtained results from general bounded regular domains to smooth,compact,and bounded Riemannian manifold.The fifth chapter is about an application of the schauder estimates.It mainly uses the estimates obtained in the previous chapters to prove the existence of the solution of the simplest operator(-Δ)m,and then by the continuity method,we obtain the existence of solutions for linear parabolic differential equations of 2m order with strong ellipses in the main part.