On some regularity results of Jacobian determinants and applications

Abstract

The starting point in the theory of the regularity of the Jacobian is the celebrated result due to S. Muller [M] that for an orientation preserving mapping f e W1,n loc (Ω, Rn), J belongs to the Zygmund space L log L(K) for any compact K C Ω. The improved integrability property of the Jacobian could be observed in Orlicz-Sobolev near W-1,n loc (Ω, Rn)(see [IS], [BFS], [MO], [GIOV], [GIM])). As suggested in [BFS] it is interesting to study the regularity of the Jacobian when Df belongs to Lorentz spaces. It appears that to get positive results one cannot rely only on these but it is forced to encode the theory in the Lorentz-Zygmund spaces. It is also possible to extend this study to the couple (B,E), B : Ω → Rn, E → Rn, of vector fields on Ω, such that divB = 0 and curl E = 0, having the scalar product ‹ B,E › nonnegative. In this case we obtain results of higher integrability for the scalar product ‹ B,E ›. R. Coifman, P.L. Lions, Y. Meyer and S. Semmes in a famous paper Compensated compactness and Hardy spaces studied the regularity of the mappings with Jacobian of arbitrary sign and as a consequence of couple (B,E), belonging to Lebesgue spaces, where divB = 0 and curl E = 0 whose scalar product is of arbitrary sign. Following this idea, we study analogous regularity properties of couple in the framework of Lorentz spaces. The last chapter is devoted to study nondivergence elliptic equations, applying the results found previously. We develop a theory for elliptic equations with bounded coecients having sufficiently small BMO-norm and we find a higher integrability of the solution. More delicate is the case of unbounded coecients, our main result is a L2 log L estimate for |V2u|.