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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

There exists

Assume that

(Natural growth condition). There exist constants

for all

Further hypothesis (H3) deduces, writing

There exist

Note that we trivially have

If the domain we consider is an upper half unit ball

There exist

Here we write

By a weak solution of (

Under such assumptions, even the boundary data is smooth, one cannot expect full regularity of (

After the partial regularity results of the type in this paper were proved by Giusti and Miranda in [

Inspired by [

The technique of A-harmonic approximation [

Consider a bounded domain

Note in particular that the boundary condition (H5) means that

Combining this result with the analogous interior [

Under the assumptions of Theorem

If the domain of the main step in proving Theorem

Consider a bounded weak solution of (

Note that analogous to the above, the boundary condition

In this section we present the A-harmonic approximation lemma [

Consider fixed positive

Next we recall a slight modification of a characterization of Hölder continuous functions originally due to Campanato [

Consider

Then there exists a Hölder continuous representative of the

We close this section by a standard estimate for the solutions to homogeneous second-order elliptic systems with constant coefficients [

Consider fixed positive

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

Next we note that the Poincaré inequality in this setting for

Finally, we fix an exponent

Then we establish an appropriate inequality for Caccioppoli.

Let

Consider a cutoff function

Using (H1), (H4), (H5), and Young's inequality and noting that

In this section we proceed to the proof of the partial regularity result.

Consider

Using (

Applying in turn Young's inequality, (H3), the Caccioppoli inequality (Theorem

Multiplying (

Consider

We now set

We now set

For

For

Using (

For the time being, we restrict to the case that

Note that fix

Combining these estimates with (

Choose

We can see from (

We now choose

For any

Then we can calculate

Then we have

The next step is to go from a discrete to a continuous version of the decay estimate. Given

Combining the boundary and interior estimates [

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).

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