The Method Of A-Harmonic Approximation And Optimal Interior Partial Regularity For Nonlinear Elliptic Systems Under Controllable Growth Conditions

In this paper, we are concerned with the partial regularity for the weak solution of nonlinear elliptic system of the following type:In most direct proof of partial regularity, one uses the method of freezes the coefficients" to obtain the desired result, where used the complex and long reverse-Holder inequality or the Gehring Lemma. In this paper, we are adopted to the method of A-harmonic approximation—which was first carried out by Duzaar and Grotowski, they consider the interior partial regularity of the weak solutions of nonlinear elliptic systems with natural growth condition. The new method not only means we do not need the reverse-Holder inequality or the Gehring Lemma, but also simplifies the procedure of proof. In particular, the Holder exponent we obtained is optimal. That it is, we have the following main result:Theorem 1.1 Let u ∈ H^1,2(Ω,R^N) be a weak solution to (1.1) under the structure assumptions (A1)- (A3) and (B). Then Ω₀ is open in Ω, and u ∈ C^1,β(Ω₀, R^N). Further Ω - Ω₀ (?) ∑₁ ∪ ∑₂, whereandand in particular, Lⁿ(Ω - Ω₀) = 0.