Article 14

Volume 1, Issue 2, 2000

Article 14

REGULARITY RESULTS FOR VECTOR FIELDS OF BOUNDED DISTORTION AND APPLICATIONS

DIPARTIMENTO DI COSTRUZIONI E METODI MATEMATICI IN ARCHITETTURA
VIA MONTEOLIVETO,3
-80134~NAPOLI, ITALY
E-Mail: fiorenza@ds.unina.it

DIPARTIMENTO DI MATEMATICA E APPLICAZIONI
"R. CACCIOPPOLI"
VIA CINTIA, 80126~NAPOLI, ITALY
E-Mail: giannett@unina.it

Received 17 January, 2000; accepted 4 April 2000.
Communicated by: S. Saitoh

ABSTRACT. In this paper we prove higher integrability results for vector fields

, $(B,E)\in L^{2-\epsilon}(\Omega , \real^n) \times L^{2-\epsilon }(\Omega , \real^n),~\epsilon$ small, such that $\text{div ~}B=0$ , $\text{curl~} E=0$ satisfying a "reverse" inequality of the type

$\begin{displaymath}\vert B\vert^{2}+\vert E\vert^{2}\le \left(K+\displaystyle{\frac{1}{K}}\right)\langle B,E \rangle +\vert F\vert^{2}\end{displaymath}$

with $K\ge 1$ and $F\in L^r(\Omega , \real^n),~r>2-\epsilon$ . 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

$\begin{displaymath}\text{div ~}a(x,\nabla u)= \text{div ~}F\end{displaymath}$

If $\vert a(x,z)\vert^{2}+\vert z\vert^{2}\le \left(K+\displaystyle{\frac{1}{K}}\right) \langle a(x,z),z\rangle$ , in the homogeneous case, our method provides a new proof of the regularity result

$\begin{displaymath}u\in W^{1,2-\epsilon }_{loc}(\Omega )\Rightarrow u\in W^{1,2+\epsilon }_{loc}(\Omega )\end{displaymath}$

where $\epsilon$ is sufficiently small. A result of higher integrability for functions verifying a reverse integral inequality is used, and its optimality is proved.

Key words:
Reverse Inequalities, Finite Distortion Vector Fields, Div-Curl Vector Fields, Elliptic Partial Differential Equations

2000 Mathematics Subject Classification: 35J60, 26D15.

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Previous issues

Volume 1, Issue 1, 2000