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:
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.
We also note here that Poincaré’s inequality in this setting yields
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 positiveand (depending only onn, N, λ, L, andγ) with the property that
A standard covering argument  allows us to obtain the following.
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 ballwhich satisfies the structural conditions (H1)-(H4) and (H5)′. Then there exist positiveand (depending only onn, N, λ, L, andγ) with the property that
2 The A-harmonic approximation technique
Lemma 2.1 (A-harmonic approximation lemma)
Next we recall a characterization of Hölder continuous functions with a slight modification .
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, andwith. Then there existsdepending only onn, N, λ, andL (without loss of generality we take) such that forsatisfying (2.1) and (2.2), anyA-harmonic functionhonwithsatisfies
3 Caccioppoli inequality
In this section we prove Caccioppoli’s inequality.
Theorem 3.1 (Caccioppoli inequality)
Using conditions (H1), (H4), (H5), and the definition of weak solutions (1.2), we have
Using Young’s inequality again,
Similarly, we have
Combining these estimates in (3.3), we have
Using (H2), we thus have
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.
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
and we observe from the definition of γ, (4.6) means that
Further we note that
Using (4.11) and (4.12) we observe
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 :
Using Sobolev’s, Caccioppoli’s, and Young’s inequalities together with (4.14), we have
Combining these estimates with (4.15) and (4.16), we have
Thus from (4.17) we can see
Using these estimations, we can obtain
Thus we have
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
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).