Regularities Of Quasilinear Elliptic Equations And Systems With Discontinuous Coefficients

This dissertation mainly study the following contents:1.To establish the partial Holder continuity estimates with optimal Holder exponent for weak solutions of the divergence quasi-linear elliptic systems with VMO discontinuous coefficients;2.To study the BMO regularity for the gradients of weak solutions to degenerate elliptic A-harmonic type systems under weak conditions;3.To obtain the regularity estimates in Morrey space for the gradients of weak solutions to sub-elliptic systems with VMO discontinuous coefficients in Carnot group;4.Under the natural growth condition,s-tudying the interior Holder continuity with optimal Holder exponent for weak solutions to semi-linear sub-elliptic equations and general sub-elliptic A-harmonic equations,re-spectively.The dissertation is summarized as follows:Chapter 1 briefly describes the research background,summarizes the literatures and gives some basic concepts and facts.In Chapter 2,we study the partial Holder continuity with optimal exponent of weak solutions to second order divergence quasi-linear elliptic systems with VMO discontin-uous coefficients under the controllable growth and the natural growth,respectively.The method taken here is the modified A-harmonic approximation technique,by which the approximation relation between the weak solution to the studied systems and some A-harmonic function could be established.Connecting with Caccioppoli inequality a-gain,we could obtain the Holder continuity under "small energy"(partial regularity).Compared with the classical perturbation method,this method avoids the use of reverse Holder inequality and simplifies the proof in some sense.Chapter 3 is devoted to studying the global BMO regularity of the gradients of weak solutions to a class of degenerate elliptic systems with weaker regularity coef-ficients.Based on the generalized Morrey space estimates for the gradients of weak solutions to the degenerate elliptic systems,we established the regularity in BMO s-pace for the gradients of weak solutions.In Chapter 4,we studied the Morrey space regularity for the gradients of weak solutions to A-harmonic type elliptic systems with VMO coefficients under the control-lable growth condition while p is near to 2 and obtained the Holder continuity result with optimal exponent when Q-n<?<p.It should be pointed out that for general p,the regularity is unknown even for p-Laplacian.The method was based on reverse Holder inequality,by which the higher integrability for the gradients of weak solutions was proved.Then the desired result was obtained by the iteration inequality.Chapter 5 studied the interior Holder continuity of the bounded weak solutions to the semi-linear subelliptic equations under the natural growth condition.After lin-earization and through the upper and lower solution of the linear problems,we could get the Harnack inequality and then establish the interior Holder estimates of the weak solutions to the equation by the classical De Giorgi-Moser-Nash iteration combining the Poincare inequality and density lemma of vector fields.In Chapter 6,the interior Holder estimate for weak solution to more general A-harmonic type sub-elliptic equation under the natural growth was investigated.Based on a density lemma and the De Giorgi-Moser-Nash iteration,we obtained the local Holder continuity of each bounded solution of A-harmonic type equations.Chapter 7 is summary and some future research issues.