Hessian Estimates To Strong Solutions For Nondivergence Linear Elliptic Equations

In this dissertation,we are mainly devoted to the regularities in the Orlicz spaces and weighted Lorentz spaces for the following two problems concerning nondivergence linear elliptic equations,respectively.The first one is to show an interior regularity of Hessian estimates in the Orlicz spaces for any strong solutions for nondivergence linear elliptic equations with small partially BMO coefficients.The second one is to get a global estimate in weighted Lorentz spaces for the Dirichlet problems of nondivergence linear elliptic equations with small BMO coefficients defined in C1,1-smooth domain.In the first chapter,we briefly introduce the background and the development settings concerning our study in this dissertation.Additionally,we present some definitions in-volving the Orlicz spaces and the weighted Lorentz spaces,and recall basic lemmas with respect to the boundedness of Hardy-Littlewood maximal function and modified Vitali covering lemma.In the second chapter,we consider the following nondivergence linear elliptic equa-tions:where the leading coefficients aij are assumed to be measurable in one variable and have small BMO-norms in the other variables.A local version of Hessian estimates in Orlicz spaces to any strong solutions is established as follows:where the positive constant c is independent of f and u,? presents the Young function satisfying the condition of ?2 ? ?2,and C?(x)indicates(x1-?,x1 + ?)× B'?(x')in ?.To attain our aim,an approximation argument about the limiting problem with coefficients locally depending only on x1,the Orlicz boundedness of the Hardy-Littlewood maximal functions and an equivalent representation of Orlicz norm are employed.Furthermore,combining an extension argument of even or odd reflection,we obtain the Orlicz estimate of the flat boundary.In the third chapter,we mainly study the following Dirichlet problem of nondiver-gence linear elliptic equations defined in a bounded C1,1 smooth domain:where the leading coefficient aij is assumed to be small BMO and the weight function?belongs to Aq/2.In accordance with the Lorentz boundedness of Hardy-Littlewood max-imal function and modified Vitali covering lemma,we obtain interior and flat boundary estimates of Hessian to strong solution of the above-mentioned Dirichlet problem,respec-tively.Finally,we obtain the global Lorentz estimate by way of flattening out the boundary and finite covering lemma as follows:?u?L?q,1(?)+?Du?L?q,1(?)+?D2u?L?q,1(?)?c?f?L?q,1(?),in which the positive constant c is independent of f and u.