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

In this paper we consider regularity of the weak solution of the nonlinearelliptic systems of divergence form with Dini continuous coefficients under thecontrollable growth condition and natural growth condition:In most direct proof of partial regularity, one uses the method of "freezes thecoefficients" to obtain the disired result, which used the complex and long reverse-H(o|¨)ler inequality or the Gehring Lemma. Here we are adopted to the method ofΓ-harmonic approximation-which was first carried out by Duzaar and Grotowski.They consider the interior partial regularity of the weak solutions of nonlinearelliptic systems with H(o|¨)lder continuous coefficients. The new method simplifiesthe procedure of the proof. In particular, we get the relatively satisfying partialregularity and optimal interior partial regularity.In this paper, we use a new method -the method ofΓ-harmonic approxima-tion, to consider partial regularity theory for weak solutions of nonlinear partialdifferential systems under controllable growth condition and natural growth con-dition, respectively.Γ-harmonic approximation lemma -the key ingredient ofthe new method -puts up a bridge betweenΓ-harmonic function and nonlinearpartial differential systems, which makes us can construct a specified functioncorresponding with weak solutions u. TheΓ-harmonic approximation lemma-re-veals that there exists aΓ-harmonic function closing to the specified function inL². Making full use of those known properties ofΓ-harmonic function, one canderive the desired decay estimate and then obtain the partial regularity results. The most important new ideas in this paper is that (1) here we weaken theassumption on A with Dini continuity instead of H(o|¨)lder continuity.(2) Throughout the paper, m stands for the growth exponent of the derivationof weak solutions. We let m = 2 under the controllable growth condition. Asyou know, there is no better results on the partial regularity theory of partialdifferential systems under the controllable growth condition. In this paper, wededuce Caccioppoli second inequality under controllable growth condition by anew method.(3) In this paper, we let m≥2 under the natural growth condition. Becausethe method ofΓ-harmonic approximation can only deal with the case m≡2, andis absolutely helpless for the case m≥2. Therefore, we adopt a new method ofinterpolation technique to overcome the difficulty.The following are the main results:Theorem 2．1 Let u∈H^1,2(Ω,R^N)be a weak solution of(2．2)under theassumptions(A₁)-(A'₄),(μ₁)-(μ₃)．Then there exists a relatively closed setSingu (?)Ωsuch that u∈C¹(Ω＼Singu,R^N)．Furthermore,Singu (?)∑₁∪∑₂,whereand in particular,Lⁿ(Singu)=0．Theorem 2．2 Let u∈H^1,m(Ω,R^N)∩L^∞(Ω,R^N)be a weak solution of(2．2)under the assumptions(A₁)-(A₄),(μ₁)-(μ₃)with sup_Ω|u|=M and2aM<λ．Then there exists a relatively closed set Singu (?)Ωit such that u∈C¹(Ω＼Singu,R^N)．Furthermore Singu (?)∑₁∪∑₂,whereand in particular,Lⁿ(Singu)=0．...