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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 676215, 8 pages

http://dx.doi.org/10.1155/2013/676215

## Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

^{1}Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou, Fujian 363000, China^{2}School of Mathematical Science, Xiamen University, Xiamen, Fujian 361005, China

Received 31 December 2012; Accepted 2 April 2013

Academic Editor: Paul Eloe

Copyright © 2013 Shuhong Chen and Zhong Tan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under natural growth condition with super quadratic growth and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

#### 1. Introduction

This paper considers boundary regularity for weak solutions of quasilinear elliptic systems where is a bounded domain in with boundary of class and takes value in . Each maps into , and each maps into . A partial regularity theory of (1) must have a priori existence weak solutions. Here we assume that weak solutions exist and consider partial regularity of weak solutions directly. We further impose certain structural conditions on and with as follows.(H1) There exists such that (H2) is uniformly strongly elliptic; that is, for some we have (H3) Assume that and further that is uniformly continuous on sets of the form , for any fixed .(H4) (Natural growth condition). There exist constants and , with possibly depending on , such that for all with and .

Further hypothesis (H3) deduces, writing for , the existence of a monotone nondecreasing concave function with , continuous at 0, such that for all with [1].(H5) There exist with and a function , such that

Note that we trivially have . Further, by the Sobolev embedding theorem we have for any . If , we will take on .

If the domain we consider is an upper half unit ball , the boundary condition becomes as follows. (H5) There exist with and a function , such that

Here we write , and further , . Similarly we denote upper half balls as follows: for , we write for and set , . For we further write for and set .

*Definition 1. *By a weak solution of (1) one means a vector valued function such that
holds for all test-functions and, by approximation, for all .

Under such assumptions, even the boundary data is smooth, one cannot expect full regularity of (1) at the boundary [2]. Then, our goal is to establish partial boundary regularity.

After the partial regularity results of the type in this paper were proved by Giusti and Miranda in [3], there are some previous partial regularity results for quasilinear systems. For example, regularity up to boundary for nonlinear and quasilinear systems [4–6] has been studied by Arkhipova. Wiegner [7] established boundary regularity for systems in diagonal form first, and the proof was generalized and extended by Hildebrandt and Widman [8]. Jost and Meier [9] deduced full regularity in a neighborhood of the boundary for minima of functionals with the form . Furthermore, Duzaar et al. obtained the boundary Hausdorff dimension on the singular sets of solutions to even more general systems in [10, 11] recently. Further discussion for regularity theory can be seen in [12, 13] and their references.

Inspired by [14], in this paper, we would establish boundary regularity for quasilinear systems under natural growth condition by the method of A-harmonic approximation.

The technique of A-harmonic approximation [15–17] is a natural extension of the harmonic approximation technique, which originated from Simon's proof of Allard's [18] -regularity theorem. In this context, using the A-harmonic approximation technique, we obtain the following regularity results.

Theorem 2. *Consider a bounded domain in , with boundary of class . Let be a bounded weak solution of (1) satisfying the boundary condition (H5), and with , where the structure conditions (H1)–(H3) hold for and (H4) holds for . Consider a fixed . Then there exist positive and (depending only on , and ) with the property that
**
for some for a given implies .*

Note in particular that the boundary condition (H5) means that makes sense: in fact, we have . For , , we set . In particular, for , , we write .

Combining this result with the analogous interior [19] and a standard covering argument allows us to obtain the following bound on the size of the singular set.

Corollary 3. *Under the assumptions of Theorem 2 the singular set of the weak solution has -dimensional Hausdorff measure zero in .*

If the domain of the main step in proving Theorem 2 is a half ball, the result then is the following.

Theorem 4. *Consider a bounded weak solution of (1) on the upper half unit ball which satisfies the boundary condition and with , where the structure conditions (H1)–(H3) hold for and (H4) holds for . Then there exist positive and (depending only on , and ) with the property that**
for some for a given , implies that there holds: .*

Note that analogous to the above, the boundary condition ensures that exists, and we have indeed .

#### 2. The A-Harmonic Approximation Technique

In this section we present the A-harmonic approximation lemma [14] and some standard results due to Companato [20].

Lemma 5 (A-harmonic approximation lemma). *Consider fixed positive and , and with . Then for any given there exists with the following property: for any satisfying
**
for any (for some ) satisfying
**
for all , there exists an A-harmonic function
**
with
*

Next we recall a slight modification of a characterization of Hölder continuous functions originally due to Campanato [21].

Lemma 6. *Consider , and . Suppose that there are positive constants and , with such that, for some , there holds the following:
**
for all and ; and
**
for all and .**Then there exists a Hölder continuous representative of the -class of on , and for this representative there holds
**
for all , for a constant depending only on and .*

We close this section by a standard estimate for the solutions to homogeneous second-order elliptic systems with constant coefficients [20].

Lemma 7. *Consider fixed positive and , and with . Then there exists depending only on , , , and (without loss of generality we take ) such that, for satisfying (11), any A-harmonic function on with satisfies
*

#### 3. The Caccioppoli Inequality

In this section we would prove a suitable Caccioppoli inequality. First of all we recall two useful inequalities. The first is the Sobolev embedding theorem which yields the existence of a constant depending only on , , and such that for there holds Obviously, the inequality remains true if we replace by , which we will henceforth abbreviate simply as .

Next we note that the Poincaré inequality in this setting for yields for a constant which depends only on .

Finally, we fix an exponent as follows: if , can be chosen arbitrarily (but henceforth fixed); otherwise we take fixed in .

Then we establish an appropriate inequality for Caccioppoli.

Theorem 8 (Caccioppoli’s inequality). *Let with and be a weak solution of systems (1) under assumption conditions (H1)–(H5). Then there exists such that, for all , with , , there holds
**
where depends only on , , and and depends on these quantities, and in addition to , depends on , , , , , and .*

*Proof. *Consider a cutoff function , satisfying on and . Then the function is in and thus can be taken as a test-function.

Using (H1), (H4), (H5), and Young's inequality and noting that , we can get from (8) with positive but arbitrary (to be fixed later)
Using (H2), (19), and (20), we thus have
Thus, we fix small enough to yield the desired inequality.

#### 4. The Proof of the Main Theorem

In this section we proceed to the proof of the partial regularity result.

Lemma 9. *Consider to be a weak solution of (1), and , , for , and with . We have
**
Here and hereafter, we define
**
for .*

*Proof. *Using (8) we have

Applying in turn Young's inequality, (H3), the Caccioppoli inequality (Theorem 8), and Jensen's inequality, we calculate from (26)
where , and , for , . We introduce the notation
and further write for . For arbitrary we thus have, by rescalling,

Multiplying (29) through by yields
for defined by .

Lemma 10. *Consider satisfying the conditions of Theorem 2 and fixed; then we can find and together, with positive constants such that the smallness conditions: and together, imply the growth condition
*

*Proof. *We now set , using in turn (H1), Young's inequality, and Hölder's inequality. We have from (30)
for .

We now set , for . From (32) we then have
and from (32) we observe from the definition of (recalling also the definition of )
Further we note

For we take to be the corresponding from the A-harmonic approximation lemma. Suppose that we could ensure that the smallness condition
holds. Then in view of (33), (34), and (35) we would be able to apply Lemma 5 to conclude the existence of a function which is -harmonic, with such that

For arbitrary (to be fixed later), we have from the Campanato theorem, noting (37) and recalling also that ,

Using (38) and (39) we observe
and, hence, on multiplying this through by , we obtain the estimate

For the time being, we restrict to the case that does not vanish identically. Recalling that , using in turn Poincaré’s, Sobolev’s, and then Hölder’s inequalities, and noting also that , thus from (41) we get
for , and provided , we have

Note that fix , which is also fixed . Since , we see from the definition of
and further

Combining these estimates with (43), we can get

Choose sufficiently small that there holds: .

We can see from (46)

We now choose such that and define by
Suppose that we have
for some , where .

For any we use the Sobolev inequality to calculate

Then we can calculate

Then we have
which means that the condition (49) is sufficient to guarantee the smallness condition (37) for , for all . We can thus conclude that (46) holds in this situation. From (46) we thus have
meaning that we can apply (46) on as well, yielding
and inductively

The next step is to go from a discrete to a continuous version of the decay estimate. Given , we can find such that . Firstly we use the Sobolev inequality, to see
which allows us to deduce
and, hence,
for . Combining this with (55) and (51), we have
and more particularly
for . Recall that this estimate is valid for all and with ; assume only the condition (49) on . This yields after replacing with the boundary estimate (15) which requires to apply Lemma 6.

Combining the boundary and interior estimates [19] we can derive the desired result. As the argument for combining the boundary and interior regularity results is relatively standard, we omit it. Hence we can apply Lemma 6 and conclude the desired Hölder continuity.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11201415, 11271305), Natural Science Foundation of Fujian Province (2012J01027), and Training Programme Foundation for Excellent Youth Researching Talents of Fujian's Universities (JA12205).

#### References

- M. Giaquinta and G. Modica, “Partial regularity of minimizers of quasiconvex integrals,”
*Annales de l'institut Henri Poincaré*, vol. 3, pp. 185–208, 1986. View at Google Scholar - M. Giaquinta, “A counter-example to the boundary regularity of solutions to elliptic quasilinear systems,”
*Manuscripta Mathematica*, vol. 24, no. 2, pp. 217–220, 1978. View at Publisher · View at Google Scholar · View at Scopus - E. Giusti and M. Miranda, “Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari,”
*Archive for Rational Mechanics and Analysis*, vol. 31, no. 3, pp. 173–184, 1968. View at Publisher · View at Google Scholar · View at Scopus - A. A. Arkhipova, “Regularity results for quasilinear elliptic systems with nonlinear boundary conditions,”
*Journal of Mathematical Sciences*, vol. 77, no. 4, pp. 3277–3294, 1995. View at Publisher · View at Google Scholar · View at Scopus - A. A. Arkhipova, “On the regularity of solutions of boundary-value problem for quasilinear elliptic systems with quadratic nonlinearity,”
*Journal of Mathematical Sciences*, vol. 80, no. 6, pp. 2208–2225, 1996. View at Google Scholar · View at Scopus - A. A. Arkhipova, “On the regularity of the oblique derivative problem for quasilinear elliptic systems,”
*Journal of Mathematical Sciences*, vol. 84, no. 1, pp. 817–822, 1997. View at Google Scholar · View at Scopus - M. Wiegner, “A-Priori Schranken für Lösungen gewisser elliptischer Systeme,”
*Manuscripta Mathematica*, vol. 18, no. 3, pp. 279–297, 1976. View at Publisher · View at Google Scholar · View at Scopus - S. Hildebrandt and K. O. Widman, “On the Hölder continuity of weak solutions of quasilinear elliptic systems of second order,”
*Annali della Scuola Normale Superiore di Pisa IV*, vol. 4, pp. 144–178, 1977. View at Google Scholar - J. Jost and M. Meier, “Boundary regularity for minima of certain quadratic functionals,”
*Mathematische Annalen*, vol. 262, no. 4, pp. 549–561, 1983. View at Publisher · View at Google Scholar · View at Scopus - F. Duzaar, J. Kristensen, and G. Mingione, “The existence of regular boundary points for non-linear elliptic systems: to the memory of Sergio Campanato,”
*Journal fur die Reine und Angewandte Mathematik*, vol. 2007, no. 602, pp. 17–58, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Kristensen and G. Mingione, “Boundary regularity in variational problems,”
*Archive for Rational Mechanics and Analysis*, vol. 198, no. 2, pp. 369–455, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Hildebrandt, “Nonlinear elliptic systems and harmonic mappings,” in
*Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations*, vol. 1, pp. 481–615, Science Press, Beijing, China, 1982, See also: Vorlesungsreihe 3, SFB 72, UniversitÄat, Bonn, Germany, 1980. - S. Hildebrandt, “Quasilinear elliptic systems in diagonal form,” in
*Systems of Nonlinear Partial Differential Equations*, J. M. Ball, Ed., pp. 173–217, Proceedings of Nato Advanced Study Institute, Oxford, UK, 1983. View at Google Scholar - J. F. Grotowski, “Boundary regularity for quasilinear elliptic systems,”
*Communications in Partial Differential Equations*, vol. 27, no. 11-12, pp. 2491–2512, 2002. View at Publisher · View at Google Scholar · View at Scopus - F. Duzaar and K. Steffen, “Optimal interior and boundary regularity for almost minimal currents to elliptic integrands,”
*Journal für die Reine und Angewandte Mathematik*, vol. 546, pp. 73–138, 2002. View at Google Scholar - F. Duzaar and J. F. Grotowski, “Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation,”
*Manuscripta Mathematica*, vol. 103, no. 3, pp. 267–298, 2000. View at Google Scholar · View at Scopus - J. F. Grotowski, “Boundary regularity for nonlinear elliptic systems,”
*Calculus of Variations and Partial Differential Equations*, vol. 15, no. 3, pp. 353–388, 2002. View at Google Scholar - W. K. Allard, “On the first variation of a varifold,”
*Annals of Mathematics*, vol. 95, pp. 225–254, 1972. View at Google Scholar - M. Giaquinta,
*Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems*, Princeton University Press, Princeton, NJ, USA, 1983. - S. Campanato, “Equazioni ellittiche del II
^{o}ordine e spazi ${L}^{2\text{,}\lambda}$,”*Annali di Matematica Pura ed Applicata*, vol. 69, no. 1, pp. 321–381, 1965. View at Publisher · View at Google Scholar · View at Scopus - S. Campanato, “Proprietà di Hölderianità di alcune classi di funzioni,”
*Annali della Scuola Normale Superiore di Pisa Series III*, vol. 17, pp. 175–188, 1963. View at Google Scholar