Partial Regulatiry For Weak Solutions Of Nonlinear Elliptic Systems With Dini Continuous Coefficients

In this paper, we consider regularity of the weak solution of the nonlinear elliptic systems of divergence form with Dini continuous coefficients under the controllable growth condition and natural growth condition:In most direct proof of partial regularity, one uses the method of "freezes the coefficients" to obtain the disired result.Precisely,by "freezes the coefficients", we obtain an elliptic system with constant coefficients, and the solution of the Dirichlet problem associated to these coefficients with boundary date u and the solution itself canthenbe compared. Then we can obtain the important decay estimate by iterating and yielding the results of partial regularity. The complex and long reverse-H(?)lder inequality or the Gehring Lemma is needed in this procedure.What makes things worse is that the Holder exponent of partial regularity obtained by this method is not optimal. It means that one just can get a H(?)lder exponent more litter than the one in the H(?)lder continuity condition of the given coefficient function. Here, we adopt the method of A-harmonic approximation-which was first carried out by Duzaar and Grotowski. They considered the interior partial regularity of the weak solutions of nonlinear elliptic systems with Holder continuous coefficients. The new method simplifies the procedure of the proof. In particular, we get the relatively satisfying partial regularity and optimal interior partial regularity.Also, we use a new method-the method of A-harmonic approximation, to consider partial regularity theory for weak solutions of nonlinear partial differen- tial systems under controllable growth condition and natural growth condition, respectively.A-harmonic approximation lemma ?the key ingredient of the new method-sets up a bridge between A-harmonic function and nonlinear partial differential systems, which makes us can construct a specified function corresponding with weak solutions u.The A-harmonic approximation lemma reveals that there is a A-harmonic function closing to the specified function in L².In order to making full use of those known properties of A-harmonic function, one can derive the desired decay estimate and then obtain the partial regularity results.Throughout the paper, we denote by m the growth exponent of the derivation of weak solutions. When m=2 under the natural growth condition, as you know, there is no better results on the partial regularity theory of partial differential systems under growth condition.we deduce Caccioppoli second inequality under the natural growth condition by a new method.In the case 12+|ξ-(?)|²)^?)(1+|p|), where x,(?)∈Ω,ξ,(?)∈R^N,p∈R^nN,we have Theorem 2.1:Let u∈H^1,2(Ω,R^N) be a weak solution of (2.2') under the assumptions (H₁)-(H₃),(μ₁)-(μ₃).Then there exists a relatively closed set Singu(?)Ωsuch that u∈C¹(Ω\Singu,R^N).Furthermore, Singu(?)Σ₁∪Σ₂,whereand in particular,Lⁿ(Singu)=0.(2) When 1m+ |ξ-(?)|^m)(1+|p|)^?,where x,(?)∈Ω,ξ,(?)∈R^N,p∈R^nN,we have Theorem 5.1:Let u∈W^1,m(Ω,R^N)∩L^∞(Ω,R^N) be a weak solution of (2.2) under the assumptions (H₁)-(H₄),(μ₁)-(μ₂) with sup_Ω|u|=M.Then there exists a relatively closed set Singu(?)Ωit such that u∈C¹(Ω\Singu,R^N).Furthermore Singu(?)Σ₁∪Σ₂,whereand in particular,Lⁿ(Singu)=0.