The Regularity Research For Elliptic Equations On Non-smooth Domains

In this dissertation, we will consider the elliptic euqation by two part. In the first part,we discussed the Dirichlet problem of second order elliptic equation with divergence form. Andobtained the Hp boundness of solution$where Lu = ?div(A?u) + V u, Here we consideredtwo cases, first ? is a C2 domain, second is a Lipschitz domainaWhile, In the second part,we have studied the local boundness and Ho¨lder continuity for P-Laplacian and second orderquasi-linear elliptic equation with with Divergence form. This paper has five chapters.In the first chapter, we main obtained the important theory in the regularity of the solu-tions of Schro¨dinger equatiion, which is L2 theory. Our method as follows, first we obtainedsome impotant estimates for Green function which related operate L on ?, then using integtalrepresentation of solution which is double layer potential theory.In the second chapter, which devided by two scetions, we proved that when ? are C2domains and Lipschitz domains respectively, Dirichlet problem (1.1.1) has unique solution, andsencond order derivative satifies Hp boundness.In the third chapter, we turn our attention to another orientation which is local boundnessand H¨older continuity for non-linear elliptic equation. Here we study P-Laplacian equation??pu(x) + V (x)|u(x)|p?2u(x) = f(x). where ?pu =: div(|?u|p?2?u), V (x) is a nonnegativepotential which satify some given conditons, In this chapter we obtained |u(x)?u(y)| C(L)|x?y|α, where C rely on some fixed constants, andα∈(0, 1).In the forth chapter, we generalized the results of P-Laplace, which study the more commonquasi-linear elliptic equation with divergence form ?divA(x, ?u) + V (x)|u(x)|p?2u(x) = f(x).where the coe?cient A(x) satifies degeneration conditions. Nonnegative potential V (x) someproper condition. We obtain the similiar result as third chapter.The final chapteris the summarization of the last three chapters. We put forward somequestions which are related to this dissertation and these questions haven't been solved. At the same time, we also give several plausible suggestions to solve these questions.