© S. Chen and Z. Tan. 2010
Received: 16 November 2009
Accepted: 18 March 2010
Published: 6 April 2010
We consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal.
where is a bounded domain in , and taking values in , and has value in . , stand for the of and . To define weak solution to (1.1), one needs to impose certain structural and regularity conditions on and the inhomogeneity , as well as to restrict to a particular class of functions as follows, for ,
Even under reasonable assumptions on and , in the case of systems (i.e., ) one cannot, in general, expect that weak solutions of (1.1) will be classical, that is, -solutions. This was first shown by De Giorgi [1, 2]. The goal, then, is to establish partial regularity theory. We refer the reader to monographs of Giaquinta [3, 4] for an extensive treatment of partial regularity theory for systems of the form (1.1), as well as more general elliptic systems.
In the class direct proofs, one "freezes the coefficients" with constant coefficients. The solution of the Dirichlet problem associated to these coefficients with boundary data and the solution itself can then be compared. This procedure was first carried out by Giaquinta and Modica .
But the technique of harmonic approximation is to show that a function which is "approximately-harmonic" lies close to some harmonic function. This technique has its origins in Simon's proof  of the regularity theorem of Allard . Which also be used in  to find a so-called -regularity theorem for energy minimizing harmonic maps. The technique of harmonic approximation allows the author to simplify the original -regularity theorem due to Schoen and Uhlenbeck .
In the remarkable proof when given by Duzaar and Grotowski in , the key difference is that the solution is compared not to the solution of the Dirichlet problem for the system with frozen coefficients, but rather to an A-harmonic function which is close to in , where is a function corresponding from weak solutions. In particular, the optimal regularity result can be obtained. In [11, 12], we deal with the optimal partial regularity of the weak solution to (1.1) for the case by the method of A-harmonic approximation technique, which is advantage to the result of . The extension of A-harmonic approximation technique also can be found in [14, 15].
and the optimal regularity. Now we may state the main result.
In this section, we present the -harmonic approximation lemma, the key ingredient in proving our regularity result, and some useful preliminaries will be need in later. At first, we introduce two new functions.
We use a number of properties of which can be found in [17, Lemma ].
The next result we would state is the -harmonic approximation lemma, which is prove in .
Then we would recall a simple consequence of the a prior estimates for solutions of linear elliptic systems of second order with constant coefficients; see [17, Proposition ] for a similar result.
The next lemma is a more general version of [17, Lemma ], which itself is an extension of [3, Lemma , Chapter V]. The proof in which can easily be adapted to the present situation by replacing the condition of homogeneity by Lemma 2.1(ii).
Lemma 2.6 (Poincare-type inequality).
We conclude the section with an algebraic fact can be retrieved again from , Lemma 2.1.
3. A Caccioppoli Second Inequality
In order to prove the main result, our first aim is to establish a suitable Caccioppoli inequality.
Lemma 3.1 (Caccioppoli second inequality).
The proof is now completed by applying Lemma 2.5.
4. The Proof of the Main Theorem
We next establish an initial excess-improvement estimate, assuming that the excess is initially sufficient small. We also define , and , where stands for the constants form Lemma 2.1(vi). The precise statement is the following.
Lemma 4.2 (excess-improvement).
The regularity result then follows from the fact that this excess-decay estimate for any in a neighborhood of . From this estimate we conclude (by Campanato's characterization of Hölder continuous functions [19, 20]) that has the modulus of continuity by a constant times . By Lemma 2.1(iv) this modulus of continuity carries over to .
This work was supported by NCETXMU and the National Natural Science Foundation of China-NSAF (no: 10976026).
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