Boundary regularity for elliptic systems with supquadratic growth
© Chen and Tan; licensee Springer 2012
Received: 5 May 2012
Accepted: 20 September 2012
Published: 3 October 2012
We consider boundary regularity for weak solutions of quasilinear elliptic systems with the supquadratic growth condition and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. By an element covering argument combined with existing interior partial regularity results, we establish the boundary regularity result.
where Ω is a bounded domain in with a boundary of class , , u and take values in , . Each maps into R, and each maps into R. In theory partial regularity to (1.1) needs a priori existence of weak solutions. In order to define weak solutions to (1.1), one needs to impose certain structural and regularity conditions on and the inhomogeneity as follows. For ,
for all , , and where for some .
Note that we trivially have . Further, by the Sobolev embedding theorem, we have for any . If , we will take on Ω.
If the domain in this case is a half unit ball, then the boundary condition is as follows:
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 , .
It is easily seen that by closure (1.2) holds for every φ in .
Under such assumptions on the structural conditions, full regularity of (1.1) at the boundary cannot, in general, be expected  even if the boundary data is smooth. Then our goal is to establish a partial boundary regularity for weak solutions of systems (1.1).
There are some previous partial regularity results for quasilinear systems. For example, Arkhipova has studied regularity up to the boundary for nonlinear and quasilinear systems [2–4]. For systems in a diagonal form, boundary regularity was first established by Wiegner , and the proof was generalized and extended by Hildebrandt-Widman . In the case of minima of functionals of the form , Jost-Meier  established full regularity in a neighborhood of the boundary.
The result which is most closely related to that given here was shown by Grotowski . Grotowski obtained boundary partial regularity results for more general systems: , . In this paper, we extend the results in . However the results in the current paper do not follow from those in ; in the current situation, we need only to impose weaker structure conditions and, at the same time, we can obtain stronger conclusions.
The technique of A-harmonic approximation is a natural extension of the technique of harmonic approximation. This technique has its origins in Simon’s  proof of the regularity theorem of Allard . The technique of A-harmonic approximation then refers to the direct analog of the above situation. The interior version of this technique has previously been applied by Duzaar and Grotowski in .
In this context, as the argument for combining the boundary and the interior estimates is relatively standard, we omit it and obtain the following results.
for some for a given implies .
Note in particular that the boundary condition (H5) means that makes sense; in fact, we have .
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 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 is then:
for some for a given implies that holds.
Note that analogously to above, the boundary condition (H5)′ ensures that exists, and we have indeed .
2 The A-harmonic approximation technique
Next, we recall a slight modification of a characterization of Hölder continuous functions originally due to Campanato .
for all and .
holds for all , for a 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
holds for , . Obviously, the inequality remains true if we replace by , which we will henceforth abbreviate simply as .
for a constant which depends only on n.
Finally, we fix an exponent as follows: if , σ can be chosen arbitrarily (but henceforth fixed); otherwise, we take σ fixed in .
Then we prove an appropriate inequality for Caccioppoli.
Theorem 3.1 (Caccioppoli inequality)
where , depending only on λ and L, m and depending on these quantities and, in addition, on , and s.
Choosing ε small enough, we can get the desired result immediately. □
4 The proof of the main theorem
In this section, we proceed with the proof of the partial regularity result, and hence consider u to be a weak solution of (1.1). We consider and , , for , and with . First of all, we have
for defined by . □
Therefore, we can find such that .
provided and where .
We then fix , note that this also fixed δ. Since , from the definition of I, we can get , and .
for some , where .
for given by . Recalling that this estimate is valid for all and ρ with , assuming only the condition (4.20) on . This yields - after replacing R by 6R - the boundary estimate (2.7) requiring to apply Lemma 2.2.
The analogous interior estimate to (4.22) is standard, and we obtain the desired interior estimate (2.8) by a standard argument. Hence, we can apply Lemma 2.2 and conclude the desired Hölder continuity. □
This work was supported by the National Natural Science Foundation of China (Nos: 11201415, 11271305); the Natural Science Foundation of Zhejiang Province (Y6110078); the Natural Science Foundation of Fujian Province (2012J01027) and Training Programme Foundation for Excellent Youth Researching Talents of Fujian’s Universities (JA12205).
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