In this section we proceed to the proof of partial regularity result.
Lemma 4.1 Consider 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 4.2 Consider u satisfying the conditions of Theorem 1.1 and σ fixed, then we can find δ and 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. □