Radice, Teresa (2006) On some regularity results of Jacobian determinants and applications. [Tesi di dottorato] (Unpublished)

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Item Type: Tesi di dottorato
Language: English
Title: On some regularity results of Jacobian determinants and applications
Creators:
CreatorsEmail
Radice, TeresaUNSPECIFIED
Date: 2006
Date Type: Publication
Number of Pages: 103
Institution: Università degli Studi di Napoli Federico II
Department: Matematica e applicazioni "Renato Caccioppoli"
PHD name: Scienze computazionali e informatiche
PHD cycle: 18
PHD Coordinator:
nameemail
Ricciardi, Luigi MariaUNSPECIFIED
Tutor:
nameemail
Moscariello, GiocondaUNSPECIFIED
Date: 2006
Number of Pages: 103
Uncontrolled Keywords: Jacobian determinants, Hardy spaces, Elliptic equations
MIUR S.S.D.: Area 01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica
Date Deposited: 31 Jul 2008
Last Modified: 30 Apr 2014 19:24
URI: http://www.fedoa.unina.it/id/eprint/1083

Abstract

The starting point in the theory of the regularity of the Jacobian is the celebrated result due to S. Müller [M] that for an orientation preserving mapping f ε 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 spaces 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 spaces 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|.

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