| In this thesis,we are devoted to regularities for generalized solutions of the Dirich-let problems to two classes of elliptic equations,respectively:The first one is to derive a global weighted Lorentz estimate and Lorentz-Morrey regularity of weak solutions to generalized Stokes systems in Reifenberg flat domains;The second one is to show a reg-ularity of very weak solutions to the Dirichlet problems for A-harmonic elliptic equations in Lebesgue spaces.More precisely,we describe our main contents as follows:In chapter one,we mainly introduce the research background and the research sta-tus involving our topic at home and abroad.Recall some basic definitions and related properties used in this dissertation.In the second chapter,for two unknown quantities with the velocity u =(u1,u2…un)and the pressure P,we consider the following zero Dirichlet problems for generalized Stokes equations:where Ω c Rn with n ≥ 2 is a bounded nonsmooth domain,F =(Fia)ni,a-1 is a given matrix-valued function.Based on using the boundedness of the Hardy-Littlewood maxi-mal operators in the weighted Loren tz spaces and the modified Vi tali covering we obtain the global estimates for the derivatives of the weak solution in weighted Lorentz spaces and Lorentz-Morrey spaces,respectively,under the assumptions that the coefficients A(x)have small BMO seminorms and Ω is a Reifenberg flat domain.In the third chapter,for θ ∈ W1,q(Ω)with q>r we consider the very weak solution u ∈θ+ W1,r0(Ω)with max｛1,p-1}<r<p<n to the Dirichlet problems of A-harmonic elliptic equationsThen,by the so-called Hodge decomposition method,we get the higher integrabilities in Lebesgue spaces for its very weak solution u in accordance with r being close to p and different cases of q,respectively. |
