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Regularity results for vector fields of bounded distortion and applications  
 
  Authors: Alberto Fiorenza, Flavia Giannetti,  
  Keywords: Reverse Inequalities, Finite Distortion Vector Fields, Div-Curl Vector Fields, Elliptic Partial Differential Equations  
  Date Received: 17/01/00  
  Date Accepted: 04/04/00  
  Subject Codes:

35J60,26D15.

  
  Editors: Saburou Saitoh,  
 
  Abstract:

In this paper we prove higher integrability results for vector fields $ B,E$, $ (B,E)in L^{2-epsilon}(Omega, real^n) times L^{2-varepsilon}(Omega, real^n), varepsilon$ small, such that $ mathrm{div}  B=0$, $ mathrm{curl}  E=0$ satisfying a ``reverse'' inequality of the type

$displaystyle vert Bvert^{2}+vert Evert^{2}le left( K+displaystyle{frac{1}{K}}right)langle B,E rangle +vert Fvert^{2}$    
with $ Kge 1$ and $ Fin L^r(Omega, real^n), r>2-varepsilon$. Applications to the theory of quasiconformal mappings and partial differential equations are given. In particular, we prove regularity results for very weak solutions of equations of the type
$displaystyle mathrm{div}  a(x,nabla u)= mathrm{div}  F.$    

If $ vert a(x,z)vert^{2}+vert zvert^{2}le left( K+1/Kright) langle a(x,z),zrangle$, in the homogeneous case, our method provides a new proof of the regularity result
$displaystyle uin W^{1,2-varepsilon}_{loc}(Omega)Rightarrow uin W^{1,2+varepsilon}_{loc}(Omega)$    
where $ varepsilon$ is sufficiently small. A result of higher integrability for functions verifying a reverse integral inequality is used, and its optimality is proved.
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