- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 676215, 8 pages
Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition
1Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou, Fujian 363000, China
2School 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.
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.
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 .(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 .
After the partial regularity results of the type in this paper were proved by Giusti and Miranda in , 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  established boundary regularity for systems in diagonal form first, and the proof was generalized and extended by Hildebrandt and Widman . Jost and Meier  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 , 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  -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  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
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 .
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 .
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)
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  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.
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).
- M. Giaquinta and G. Modica, “Partial regularity of minimizers of quasiconvex integrals,” Annales de l'institut Henri Poincaré, vol. 3, pp. 185–208, 1986.
- 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.
- 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.
- 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.
- 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.
- 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.
- M. Wiegner, “A-Priori Schranken für Lösungen gewisser elliptischer Systeme,” Manuscripta Mathematica, vol. 18, no. 3, pp. 279–297, 1976.
- 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.
- J. Jost and M. Meier, “Boundary regularity for minima of certain quadratic functionals,” Mathematische Annalen, vol. 262, no. 4, pp. 549–561, 1983.
- 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.
- J. Kristensen and G. Mingione, “Boundary regularity in variational problems,” Archive for Rational Mechanics and Analysis, vol. 198, no. 2, pp. 369–455, 2010.
- 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.
- J. F. Grotowski, “Boundary regularity for quasilinear elliptic systems,” Communications in Partial Differential Equations, vol. 27, no. 11-12, pp. 2491–2512, 2002.
- 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.
- 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.
- J. F. Grotowski, “Boundary regularity for nonlinear elliptic systems,” Calculus of Variations and Partial Differential Equations, vol. 15, no. 3, pp. 353–388, 2002.
- W. K. Allard, “On the first variation of a varifold,” Annals of Mathematics, vol. 95, pp. 225–254, 1972.
- 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 IIo ordine e spazi ,” Annali di Matematica Pura ed Applicata, vol. 69, no. 1, pp. 321–381, 1965.
- 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.