Boundary regularity for quasilinear elliptic systems with super quadratic controllable growth condition
© Chen and Tan; licensee Springer. 2014
Received: 3 November 2013
Accepted: 10 April 2014
Published: 6 May 2014
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.
Keywordsquasilinear elliptic systems controllable growth condition A-harmonic approximation technique boundary partial regularity
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.
for all , .
where if , or any exponent if ; for all , and .
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.
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.
for , . Obviously, the inequality remains true if we replace by , which we will henceforth abbreviate simply as .
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.
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 1.1 Under the assumptions of Theorem 1.1 the singular set of the weak solution u has -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.
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)
Next we recall a characterization of Hölder continuous functions with a slight modification .
for all and .
for all , with the constant depending only on n and α.
We close this section by a standard estimate for the solutions to homogeneous second order elliptic systems with constant coefficients, due originally to Campanato .
3 Caccioppoli inequality
In this section we prove Caccioppoli’s inequality.
Theorem 3.1 (Caccioppoli inequality)
where , , depend only on λ, L, m, and , and additionally on , , and also on s.
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.
here we have used the fact that and .
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.
for , .
for defined by . □
For () we have , where with . Therefore we can find such that .
We then fix , note that this also fixes δ. Since , we see from the definition of γ: , and further .
We choose small enough, such that we have .
for some , where .
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. □
This article was supported by National Natural Science Foundation of China (Nos: 11201415, 11271305); Natural Science Foundation of Fujian Province (2012J01027).
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