Optimal partial regularity of very weak solutions to nonhomogeneous A-harmonic systems

Journal of Inequalities and Applications, Jan 2017

We study partial regularity of very weak solutions to some nonhomogeneous A-harmonic systems. To obtain the reverse Hölder inequality of the gradient of a very weak solution, we construct a suitable test function by Hodge decomposition. With the aid of Gehring’s lemma, we prove that these very weak solutions are weak solutions. Further, we show that these solutions are in fact optimal Hölder continuity based on A-harmonic approximation technique.

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Optimal partial regularity of very weak solutions to nonhomogeneous A-harmonic systems

Zhao and Chen Journal of Inequalities and Applications Optimal partial regularity of very weak solutions to nonhomogeneous A-harmonic systems Qing Zhao Shuhong Chen We study partial regularity of very weak solutions to some nonhomogeneous A-harmonic systems. To obtain the reverse Hölder inequality of the gradient of a very weak solution, we construct a suitable test function by Hodge decomposition. With the aid of Gehring's lemma, we prove that these very weak solutions are weak solutions. Further, we show that these solutions are in fact optimal Hölder continuity based on A-harmonic approximation technique. nonhomogeneous A-harmonic systems; very weak solution; optimal partial regularity; Hodge decomposition; A-harmonic approximation technique 1 Introduction A-harmonic systems of the following type: – div A(x, u, Du) = f (x), where u : → RN is a vector-valued function on a bounded domain ⊂ Rn (n ≥ ), and A(x, u, ξ ) ≤ β  + |ξ | p– such that Dξ A(x, u, ξ )ζ · ζ ≥ α  + |ξ | p– |ζ | for all x ∈ , u ∈ RN , and ξ , ζ ∈ RnN ; γ p A(x, u, ξ ) – A(x˜, u˜ , ξ ) ≤ K |u| |x – x˜|p + |u – u˜ |p p  + |ξ |  A direct consequence of this result follows immediately. –  =  ∪ ,  = x ∈ Du – (Du)x,R p dx >   = x ∈ : lim sup |ux,R| + (Du)x,R = ∞ . R→+ Moreover, we have | – | = . 2 Preliminary lemmas Before proving the results, we state a few useful lemmas. The first one is a stability result of the Hodge decomposition, from which we could construct a suitable test-function concerning estimates below the natural exponent for (.). < , we have the inequality In the end of this section, we shall introduce a form of Gehring’s lemma, which plays an important role in the proof of Theorem . It implies in particular that from it higher integrability of g(x) follows. BR/(x) g(x) p dx + C∗ BR/(x) ≤ C∗ + C∗ 3 Proof of the main theorems In this section, we give a proof of partial regularity results. Consider u solving (.) on BR(x) , where we restrict  < R < R < min{, dist(x, ∂ )}. 3.1 Proof of Theorem 1 where h satisfies D(ηv) –εD(ηv) = Dφ + h, h p––εε ≤ Cr( , N )ε D(ηv) p––εε.  ≤ CCPp–ε Dv p–ε + Dv p–ε –ε In view of (.) and (.), we have h p––εε ≤ Cε Dv p––εε,  where C = Cr( , N )( + CCPp–ε ). In particular, combining (.) and (.), we find D(ηv) –εD(ηv) – h p–ε –ε D(ηv) –εD(ηv) p–ε + h p––εε –ε ≤ D(ηv) p––εε + Cr( , N )ε D(ηv) p––εε Substituting (.) into this estimate, we have Dφ p––εε ≤ C Dv p––εε,  where C = ( + Cr( , N )ε)( + CCPp–ε ). which by Lemma  yields E(η, v) ≤ ε  –+ εε |vDη|–ε. Dφ = E(η, v) + |ηDv|–εηDv – h. A(x, u, Du) · h dx – ˆ Combining this equality with (.), we find A(x, u, Du) · h dx = ˆ = – ˆ ≤ I + I + I + I, I = – I = A(x, u, Du) · h dx ; which can be deduced from (H) immediately. Then we infer that BR/(x) I ≤ ≤ β  + |p| p– ˆ ≤ β  + |p| p– ˆ ε|Du – p|p–ε + ε– p––ε  p– dx p–ε By (H) we have I ≤ A(x, u, Du) |h| dx ≤  |Du – p| I ≤  |Du – p| |Du – p|  + |Du – p + p|   + |p| + |Du – p| |Du – p| p– ˆ |h| dx + p–β  + |p|  |h| dx. |Du – p| |Du – p| |Du – p| |Du – p| |Du – p|  + |Du|   + |p| + |Du – p|  + |p| + |Du – p| I ≤ Denoting βε +ε ( C )–εp– by C, we have –ε R p–ε p–ε Using Young’s inequality with exponents –ε and p– and Poincaré’s inequality with con K ≤ |Du – p| Letting p = n(p–ε) and q = (n–p+)(–ε) , we see that  < p < ∞,  < q < ∞, and = . With the aid of Hölder’s inequality, we can estimate K ≤ |Du – p| |v| n–p+ dx Now we set p = constant Cs, we get . Then . Using the Sobolev-Poincaré inequality with K ≤ Cs ≤ Cs I ≤ C  + |p|  εCP |Du – p| + C  + |p| Finally, we estimate I. Using Hölder’s inequality with exponents I ≤ Setting p = , we have can apply the Sobolev-Poincaré inequality to get |Du – p| |Du – p| . (.) I ≤ Cs I ≤ CsC BR/(x) |Du – p| + C  + |p| |Du – p| |Du – p| |Du – p| |Du – p| |Du – p| BR/(x) Rearranging this inequality, we have |Du – p| |Du – p|  + C( + |p| )  ε   , and C = α   (β( + |p| )  ε p– + p–β( + |p|)  Cε × BR/(x) BR/(x) |Du – p|r dx BR/(x) where C = α   CCs–ε(αnRn) p–ττ–ε . n p– |Du – p|r dx + C Competing interests The authors declare that they have no competing interests. Authors’ contributions Both authors contributed equally to writing of this paper. Both authors read and approved the final manuscript. Acknowledgements The authors would like to thank the anonymous referee for careful reading the manuscript and valuable comments. This work was supported by the National Natural Science foundation of China under Grant No. 11571159. 1. 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Qing Zhao, Shuhong Chen. Optimal partial regularity of very weak solutions to nonhomogeneous A-harmonic systems, Journal of Inequalities and Applications, 2017, 23, DOI: 10.1186/s13660-017-1297-z