- Research Article
- Open Access

# Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case under Natural Growth Condition

- Shuhong Chen
^{1, 2}Email author and - Zhong Tan
^{2}

**2010**:680714

https://doi.org/10.1155/2010/680714

© S. Chen and Z. Tan. 2010

**Received:**16 November 2009**Accepted:**18 March 2010**Published:**6 April 2010

## Abstract

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.

## Keywords

- Weak Solution
- Partial Regularity
- Young Inequality
- Regularity Theorem
- Elementary Inequality

## 1. Introduction

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 ,

for all , and ; without loss of generality, we take .

or

Definition 1.1.

for all .

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 [5].

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 [6] of the regularity theorem of Allard [7]. Which also be used in [8] 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 [9].

In the remarkable proof when given by Duzaar and Grotowski in [10], 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 [13]. 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.

Theorem 1.2.

In particular, .

## 2. The -Harmonic Approximation Technique and Preliminary Lemmas

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.

for any .

We use a number of properties of which can be found in [17, Lemma ].

Lemma 2.1.

Let and be the functions defined in (2.1). Then for any and there holds:

(i)

(ii)

(iii)

(iv)

(v)

(vi) for all with

The inequalities (i)–(iii) also hold if we replace by .

The next result we would state is the -harmonic approximation lemma, which is prove in [18].

Lemma 2.2 ( -harmonic approximation lemma).

Definition 2.3.

for all .

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.

Lemma 2.4.

where the constant depends only on , and .

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 2.5.

And then we state a Poincare type inequality involving the function , which have been found in [17] and, in a sharp way, in [18].

Lemma 2.6 (Poincare-type inequality).

where . In particular, the previous inequality is valid with replaced by .

We conclude the section with an algebraic fact can be retrieved again from [16], Lemma 2.1.

Lemma 2.7.

for any , not both zero if .

## 3. A Caccioppoli Second Inequality

For , we define and we simply write .

In order to prove the main result, our first aim is to establish a suitable Caccioppoli inequality.

Lemma 3.1 (Caccioppoli second inequality).

where and the constant .

Proof.

for

The proof is now completed by applying Lemma 2.5.

## 4. The Proof of the Main Theorem

In this section we proceed to the proof of the partial regularity result and hence consider to be a weak solution of (1.1). Then we have the following.

Lemma 4.1.

for and

Proof.

here we have used for .

for .

for

where is defined in Lemma 3.1.

Combining the above of with (4.4) and noting the definition of , we can get the lemma immediately.

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).

Here one uses the abbreviate .

Proof.

where stands for the constant from Lemma 2.1(vi).

where the constant depends only on , and .

where the constant has the same dependencies as .

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 .

## Declarations

### Acknowledgments

This work was supported by NCETXMU and the National Natural Science Foundation of China-NSAF (no: 10976026).

## Authors’ Affiliations

## References

- De Giorgi E: Frontiere orientate dimisura minima.
*Seminario di Matematica della. Scuola Normale Superiore di Pisa, 1961, Pisa, Italy*1–65.Google Scholar - De Giorgi E: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico.
*Bollettino della Unione Matematica Italiana*1968, 1: 135–137.MathSciNetMATHGoogle Scholar - Giaquinta M:
*Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies*.*Volume 105*. Princeton University Press, Princeton, NJ, USA; 1983:vii+297.Google Scholar - Giaquinta M:
*Introduction to Regularity Theory for Nonlinear Elliptic Systems, Lectures in Mathematics ETH Zürich*. Birkhäuser, Berlin, Germany; 1993:viii+131.MATHGoogle Scholar - Giaquinta M, Modica G: Regularity results for some classes of higher order nonlinear elliptic systems.
*Journal für die Reine und Angewandte Mathematik*1979, 311/312: 145–169.MathSciNetMATHGoogle Scholar - Simon L:
*Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University*.*Volume 3*. Australian National University Centre for Mathematical Analysis, Canberra, Australia; 1983:vii+272.Google Scholar - Allard WK: On the first variation of a varifold.
*Annals of Mathematics*1972, 95: 417–491. 10.2307/1970868MathSciNetView ArticleMATHGoogle Scholar - Simon L:
*Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich*. Birkhäuser, Basel, Germany; 1996:viii+152.View ArticleGoogle Scholar - Schoen R, Uhlenbeck K: A regularity theory for harmonic maps.
*Journal of Differential Geometry*1982, 17(2):307–335.MathSciNetMATHGoogle Scholar - Duzaar F, Grotowski JF: Optimal interior partial regularity for nonlinear elliptic systems: the method of -harmonic approximation.
*Manuscripta Mathematica*2000, 103(3):267–298. 10.1007/s002290070007MathSciNetView ArticleMATHGoogle Scholar - Chen S, Tan Z: Optimal interior partial regularity for nonlinear elliptic systems under the natural growth condition: the method of A-harmonic approximation.
*Acta Mathematica Scientia. Series B*2007, 27(3):491–508. 10.1016/S0252-9602(07)60049-6MathSciNetView ArticleMATHGoogle Scholar - Chen S, Tan Z: The method of -harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the controllable growth condition.
*Journal of Mathematical Analysis and Applications*2007, 335(1):20–42. 10.1016/j.jmaa.2007.01.042MathSciNetView ArticleMATHGoogle Scholar - Tan Z: -partial regularity for nonlinear elliptic systems.
*Acta Mathematica Scientia. Series B*1995, 15(3):254–263.MathSciNetMATHGoogle Scholar - Chen S-H, Tan Z: The method of -harmonic approximation and optimal interior partial regularity for energy minimizing -harmonic maps under the controllable growth condition.
*Science in China. Series A*2007, 50(1):105–115.MathSciNetView ArticleMATHGoogle Scholar - Duzaar F, Mingione G: Regularity for degenerate elliptic problems via -harmonic approximation.
*Annales de l'Institut Henri Poincaré. Analyse Non Linéaire*2004, 21(5):735–766.MathSciNetView ArticleMATHGoogle Scholar - Acerbi E, Fusco N: Regularity for minimizers of nonquadratic functionals: the case .
*Journal of Mathematical Analysis and Applications*1989, 140(1):115–135. 10.1016/0022-247X(89)90098-XMathSciNetView ArticleMATHGoogle Scholar - Carozza M, Fusco N, Mingione G: Partial regularity of minimizers of quasiconvex integrals with subquadratic growth.
*Annali di Matematica Pura ed Applicata. Serie Quarta*1998, 175: 141–164. 10.1007/BF01783679MathSciNetView ArticleMATHGoogle Scholar - Duzaar F, Grotowski JF, Kronz M: Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth.
*Annali di Matematica Pura ed Applicata. Series IV*2005, 184(4):421–448. 10.1007/s10231-004-0117-5MathSciNetView ArticleMATHGoogle Scholar - Campanato S: Proprietà di una famiglia di spazi funzionali.
*Annali della Scuola Normale Superiore di Pisa. Classe di Scienze*1964, 18: 137–160.MathSciNetMATHGoogle Scholar - Campanato S: Equazioni ellittiche del II deg ordine espazi .
*Annali di Matematica Pura ed Applicata. Serie Quarta*1965, 69: 321–381. 10.1007/BF02414377MathSciNetView ArticleMATHGoogle Scholar

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