Regularity On The Solutions To Elliptic Equations And Parabolic Equations With Discontinuous Coefficients

During the last forty years a number of mathematicians have devoted to the study of local and global regularity properties to elliptic equations and parabolic equation with discontinuous coefficients. Especialy, by the aid of Calderon-Zygmund singular integral theory ,the study on the regularity of solutions has greatly advaced. In 1991, making use of Calderon-Zygmund singular integral theory ,Chiarenza, Frasca and Longo discussed the local W^2,p regularity of solutions to the nondivergence elliptic equations with VMO coefficients. They gave the derivative formula of solutions by use of the freezing coefficients method, then made use of the L^p- estimation of the singular integral and the commutator to obtain the local regularity of solutions([22]). Again in 1993, they got the global regularity of the solutions to the nondivergence elliptic equations([23]). From then on, there have been great deal of references to study the local or global regularity of solutions in L^p or Morrey spaces for the nondivergence elliptic or parabolic equations ([10, 21, 26, 31, 35, 33, 34, 37, 36, 44, 46, 51, 53, 54, 75]). In 2000, Ragusa considered the local regularity of solutions in Morrey spaces for di-vegence elliptic equations[60]. Lihe Wang [79] gave a different proof for Calderon-Zygmund estimation in 2003 by use of Vitali covering lemma, maximal function theory and compactness method. With the aid of the method, Wang and Byun studied the L^p regularity of solutions for divergence elliptic and parabolic equations with small BMO coefficients ([6, 7, 12, 13. 14]).The present paper will use two methods to discuss the regularity of solutions to the elliptic equations and parabolic equations with discontinuous coefficients, we study the global regularity in Morrey spaces and Holder regularity of solutions to the divergence elliptic equations with main coefficients belonging to VMO and with lower order terms, and study the local regularity in L^p and Morrey spaces and Holder...