Partial Regularity For Weak Solutions Of Nonlinear Elliptic Equations

This thesis mainly studies the partial regularity for weak solutions of non-linear ellip-tic equations under different structure conditions by using the classical method(freezing the coefficient method)and A-harmonic approximation.This thesis mainly contains three parts.The first part is Chapter 3,we study the partial regularity for weak solutions of non-linear subquadratic elliptic equations under controllable growth condition by using the classical method,the conclusion holds in the interior and holds up to the boundary.The second part is Chapter 4,we study the partial regularity in the interior for weak solutions of non-linear subquadratic elliptic equations with VMO-coefficients by using A-harmonic approximation,this part contains homogeneous system and inhomogeneous system under natural growth condition and controllable growth condition.The third part is Chapter 5,we study the partial regularity in the interior for weak solutions of non-linear supquadratic elliptic equations with VMO-coefficients under natural growth condition and controllable growth condition by using A-harmonic approximation.In Chapter 2,we introduce some basic notions and definitions,there are also some basic inequalities,that will be used in the subsequent chapter.In Chapter 3,we consider the partial regularity for weak solutions of non-linear subquadratic elliptic equations in divergence form under controllable growth,condition.We prove that the weak solution u is locally Holder continuous besides a singular set by using the classical method(freezing the coefficient method)and classical Morrey-type estimates.Here the Hausdorff dimension of the singular set is less than n-p.This result not only holds in the interior,but also holds up to the boundary.In Chapter 4,we consider two problems,one problem is the partial regularity for weak solutions of non-linear subquadratic homogeneity elliptic equations in divergence form with VMO-coefficients,another is the partial regularity for weak solutions of non-linear subquadratic inhomogeneity elliptic equations in divergence form with VMO-coefficients under natural growth condition and controllable growth condition.We obtain that the weak solution u is locally Holder continuous besides a singular set by using A-harmonic approximation,where the Holder exponent α ∈(0,1).The Hausdorff dimension of the singular set is less than n-p.When dealing with subquadratic case,we need to use the V-function and some of its properties.In Chapter 5,we consider the partial regularity for weak solutions of non-linear supquadratic inhomogeneity elliptic equations in divergence form with VMO-coefficients under natural growth condition and controllable growth condition.We obtain that the weak solution u is locally Holder continuous besides a singular set by using A-harmonic approximation,where the Holder exponent α ∈(0,1).