We consider the boundary regularity for weak solutions to quasilinear elliptic systems under a super quadratic controllable growth condition, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on the interior partial regularity, this result yields an upper bound on the Hausdorff dimension of a singular set at the boundary.
Keywords:quasilinear elliptic systems; controllable growth condition; A-harmonic approximation technique; boundary partial regularity
In this paper we are concerned with partial regularity for weak solutions of quasilinear elliptic systems:
where Ω is a bounded domain in , , , and u and take values in . Here each maps into R, and each maps into R. For , we have the following.
(H1) There exists such that
(H2) is uniformly strongly elliptic, that is, for some we have
(H3) There exists a monotone nondecreasing concave function with , continuous at 0, such that
for all , .
(H4) The fulfill the following controllable growth condition:
where if , or any exponent if ; for all , and .
(H5) There exist s with and a function , such that we have
Note that we trivially have . Further, by Sobolev’s embedding theorem we have for any . If , we will take on Ω.
If the domain we consider is an upper half unit ball , the boundary condition is the following.
(H5)′ There exist s with and a function , such that we have
Here we write , and further , . For we write for , and we set , . We further write , and we set , . For bounded with we denote the average of a given function by , i.e. . For , we set . In particular, for , , we write .
Now we can definite weak solutions to systems (1.1). Because there is a very large literature on the existence of weak solutions [1,2], we assume that a weak solution exists  and deal with the problem of regularity directly.
Definition 1.1 By a weak solution of (1.1) we mean a vector-valued function such that
holds for all test-functions and, by approximation, for all , where we have introduced the notation
In the current situation, Sobolev’s embedding theorem yields the existence of a constant depending only on s, n, and N such that we have
for , . Obviously, the inequality remains true if we replace by , which we will henceforth abbreviate simply as .
We also note here that Poincaré’s inequality in this setting yields
for a constant depending only on n.
Finally, we fix an exponent as follows: if , σ can be chosen arbitrary (but henceforth fixed); otherwise we take σ fixed in .
Under such assumptions, one cannot expect that weak solutions to (1.1) will be classical . This was first shown by De Giorgi . Thus, our goal is to establish a partial regularity for weak solutions of systems (1.1).
There are some previous partial regularity results at boundary for inhomogeneous quasilinear systems. Arkhipova has studied regularity up to the boundary for nonlinear and quasilinear systems [6-8]. For systems in diagonal form, boundary regularity was first established by Wiegner , and the proof was generalized and extended by Hildebrandt-Widman . Jost-Meier  established full regularity in a neighborhood of boundary for minima of functionals with the form .
The results which are most closely related to that given here were shown in  and . In this paper, we would get the desired conclusions by the method of A-harmonic approximation. The A-harmonic approximation technique is a natural extension of harmonic approximation technique. In  Simon used harmonic approximation method to simplify Allard’s  regularity theorem and later on Schoen and Uhlenbeck’s  regularity result for harmonic maps. The idea was generalized to more general linear operators by Duzaar and Steffen , in order to deal with the regularity of almost minimizers to elliptic variational integrals in the setting of geometric measure theory. As a by-product Duzaar and Grotowski  were able to use the idea of A-harmonic approximation to deal with elliptic systems under quadratic growth, even to the boundary points for nonlinear elliptic systems  and variational problems .
In this context, we use an A-harmonic approximation method to establish boundary regularity results.
Theorem 1.1Let Ω be a bounded domain in , with boundary of class . Letube a weak solution of (1.1) satisfying the structural conditions (H1)-(H5). Consider a fixed . Then there exist positive and (depending only onn, N, λ, L, andγ) with the property that
for some , , which implies .
Note in particular that the boundary condition (H5) means that makes sense: in fact, we have .
A standard covering argument  allows us to obtain the following.
Corollary 1.1Under the assumptions of Theorem 1.1 the singular set of the weak solutionuhas -dimensional Hausdorff measure zero in .
If the domain of the main step in proving Theorem 1.1 is a half ball, the result then is the following.
Theorem 1.2Consider a weak solution of (1.1) on the upper half unit ball which satisfies the structural conditions (H1)-(H4) and (H5)′. Then there exist positive and (depending only onn, N, λ, L, andγ) with the property that
for some , , which implies .
Analogously to above, the boundary condition (H5)′ ensures that exists, and .
2 The A-harmonic approximation technique
Lemma 2.1 (A-harmonic approximation lemma)
Consider fixed positiveλandL, and with . Then for any given there exists with the following property: for any satisfying
for any (for some , ) satisfying
for all , there exists anA-harmonic function
Next we recall a characterization of Hölder continuous functions with a slight modification .
Lemma 2.2Consider , , and . Suppose that there are positive constantsκandα, with such that, for some , we have the following:
for all and ; and
for all and .
Then there exists a Hölder continuous representative of the -class ofνon , and for this representative we have
for all , with the constant depending only onnandα.
We close this section by a standard estimate for the solutions to homogeneous second order elliptic systems with constant coefficients, due originally to Campanato .
Lemma 2.3Consider fixed positiveλandL, and with . Then there exists depending only onn, N, λ, andL (without loss of generality we take ) such that for satisfying (2.1) and (2.2), anyA-harmonic functionhon with satisfies
3 Caccioppoli inequality
In this section we prove Caccioppoli’s inequality.
Theorem 3.1 (Caccioppoli inequality)
Let be a weak solution of systems (1.1) under the conditions (H1)-(H5). Then there exists depending only onL, s, and , such that for all , with , , we have
where , , depend only onλ, L, m, and , and additionally on , , and also ons.
Proof Now we consider a cut off function , satisfying , on , and . Then the function is in , and thus it can be taken as a test-function.
Using conditions (H1), (H4), (H5), and the definition of weak solutions (1.2), we have
By Young’s, Hölder’s and then Sobolev’s inequalities, together with the estimate inequalities (1.4), (1.5), and the fact that , yields
Using Young’s inequality again,
here we have used the fact that and .
Similarly, we have
Combining these estimates in (3.3), we have
Noting that , we have
Using (H2), we thus have
Recalling that and , we get
Fixing ε small enough yields the desired inequality immediately. □
4 Proof of the main theorem
In this section we proceed to the proof of partial regularity result.
Lemma 4.1Consider to be a weak solution of (1.1), , , , , and with . Then
for , .
Proof From the definition of weak solution we have
Similarly as (3.4), by Hölder’s inequality and Sobolev’s embedding theorem, we have
Henceforth we restrict ρ to be sufficiently small. Applying in turn Young’s inequality, (H3), Caccioppoli’s inequality (Theorem 3.1) and then Jensen’s inequality we calculate that
For , , we introduce the notation
and further write I for . Defining the constant by , from (4.3) and Theorem 3.1, we have
For arbitrary we thus have
Multiplying through by , this yields
for defined by . □
Lemma 4.2Considerusatisfying the conditions of Theorem 1.1 andσfixed, then we can findδand together with positive constant such that the smallness conditions and together imply the growth condition
Proof Set , using in turn (H1), Young’s inequality and Hölder’s inequality, from (4.4) we can see that for :
We now set , for . From (4.5) yields
and we observe from the definition of γ, (4.6) means that
Further we note that
For we take to be the corresponding δ from the A-harmonic approximation lemma. Suppose that we could ensure that the smallness condition
holds. Then in view of (4.6), (4.7), (4.8) we would be able to apply A-harmonic approximation lemma to conclude that the existence of a function , which is -harmonic, with such that
For arbitrary (to be fixed later), from Lemma 2.3 and (4.10), recalling also that , we have
Using (4.11) and (4.12) we observe
and hence, on multiplying this through by , we obtain the estimate
For the time being, we restrict ourselves to the case that g does not vanish identically. Recalling that , (4.13) yields, using in turn Poincaré’s, Sobolev’s and then Hölder’s inequalities, noting also that :
for and provided together with , we have
For ( ) we have , where with . Therefore we can find such that .
Using Sobolev’s, Caccioppoli’s, and Young’s inequalities together with (4.14), we have
We then fix , note that this also fixes δ. Since , we see from the definition of γ: , and further .
Combining these estimates with (4.15) and (4.16), we have
We choose small enough, such that we have .
Thus from (4.17) we can see
We now choose such that , and we define by . Suppose that
for some , where .
For any we use Sobolev’s inequality to calculate
Using these estimations, we can obtain
Thus we have
which means that the condition (4.18) is sufficient to guarantee the smallness condition (4.9) for , for all . Thus, we can conclude that (4.17) holds in this situation. From (4.18) we thus have
which means that we can apply (4.18) on as well, and this yields
and inductively we find
The next step is to go from a discrete to a continuous version of the decay estimate. Given , we can find such that . Then we calculate in a similar manner to above. Firstly, we use Sobolev’s inequality (1.4) to see that
which allows us to deduce that
and hence, finally
for . We combine these estimates with (4.22) and (4.23):
and more particularly
for given by . We recall that this estimate is valid for all and ρ with , and we assume only the smallness condition (4.19) on . This yields after replacing R by 6R the boundary estimate required to apply Lemma 2.2.
Similarly, one can get the analogous interior estimate as (4.24). Applying Lemma 2.2, we can conclude the desired Hölder continuity. □
The authors declare that they have no competing interests.
SC participated in design of the study and drafted the manuscript. ZT participated in conceiving of the study and the amendment of the paper. All authors read and approved the final manuscript.
This article was supported by National Natural Science Foundation of China (Nos: 11201415, 11271305); Natural Science Foundation of Fujian Province (2012J01027).