Estimates of Jacobians by subdeterminants

Flavia Giannetti; Tadeusz Iwaniec; Jani Onninen; Anna Verde

doi:10.1007/BF02922041

Estimates of Jacobians by subdeterminants

Flavia Giannetti, Tadeusz Iwaniec, Jani Onninen, Anna Verde

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

Abstract

Let F: Ω → ℝ ⁿ be a mapping in the Sobolev space W ^1,n-1(Ω,ℝ ⁿ), n > 2. We assume that the determinant of the differential matrix DF (x) is nonnegative, while the cofactor matrix D ^#F satisfies |D ^#f|n/n 1 ε L ^P(ω), where L ^p(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L _loc ¹ (Ω). Estimates above and below L _loc ¹ (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.

Original language	English (US)
Pages (from-to)	223-254
Number of pages	32
Journal	Journal of Geometric Analysis
Volume	12
Issue number	2
DOIs	https://doi.org/10.1007/BF02922041
State	Published - 2002

Keywords

Orlicz space
cofactor matrix
distributional Jacobian
integrability of Jacobian

ASJC Scopus subject areas

Geometry and Topology

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10.1007/BF02922041

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@article{9d3d387b039b4359813af3c510dbf25a,

title = "Estimates of Jacobians by subdeterminants",

abstract = "Let F: Ω → ℝ n be a mapping in the Sobolev space W 1,n-1(Ω,ℝ n), n > 2. We assume that the determinant of the differential matrix DF (x) is nonnegative, while the cofactor matrix D #F satisfies |D #f|n/n 1 ε L P(ω), where L p(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L loc 1 (Ω). Estimates above and below L loc 1 (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.",

keywords = "Orlicz space, cofactor matrix, distributional Jacobian, integrability of Jacobian",

author = "Flavia Giannetti and Tadeusz Iwaniec and Jani Onninen and Anna Verde",

note = "Funding Information: Math Subject Classifications. primary, 42B25; secondary, 26B10. Key Words and Phrases. integrability of Jacobian, distributional Jacobian, cofactor matrix, Orlicz space. Acknowledgements and Notes. Iwaniec was supported by NSF grant DMS-0070807. This research was done while Giannetti, Onnanen and Verde were visiting Mathematics Department at Syracuse University. They wish to thank SU for the support and hospitahty.",

year = "2002",

doi = "10.1007/BF02922041",

language = "English (US)",

volume = "12",

pages = "223--254",

journal = "Journal of Geometric Analysis",

issn = "1050-6926",

publisher = "Springer New York",

number = "2",

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TY - JOUR

T1 - Estimates of Jacobians by subdeterminants

AU - Giannetti, Flavia

AU - Iwaniec, Tadeusz

AU - Onninen, Jani

AU - Verde, Anna

N1 - Funding Information: Math Subject Classifications. primary, 42B25; secondary, 26B10. Key Words and Phrases. integrability of Jacobian, distributional Jacobian, cofactor matrix, Orlicz space. Acknowledgements and Notes. Iwaniec was supported by NSF grant DMS-0070807. This research was done while Giannetti, Onnanen and Verde were visiting Mathematics Department at Syracuse University. They wish to thank SU for the support and hospitahty.

PY - 2002

Y1 - 2002

N2 - Let F: Ω → ℝ n be a mapping in the Sobolev space W 1,n-1(Ω,ℝ n), n > 2. We assume that the determinant of the differential matrix DF (x) is nonnegative, while the cofactor matrix D #F satisfies |D #f|n/n 1 ε L P(ω), where L p(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L loc 1 (Ω). Estimates above and below L loc 1 (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.

AB - Let F: Ω → ℝ n be a mapping in the Sobolev space W 1,n-1(Ω,ℝ n), n > 2. We assume that the determinant of the differential matrix DF (x) is nonnegative, while the cofactor matrix D #F satisfies |D #f|n/n 1 ε L P(ω), where L p(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L loc 1 (Ω). Estimates above and below L loc 1 (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.

KW - Orlicz space

KW - cofactor matrix

KW - distributional Jacobian

KW - integrability of Jacobian

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U2 - 10.1007/BF02922041

DO - 10.1007/BF02922041

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VL - 12

SP - 223

EP - 254

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 2

ER -

Estimates of Jacobians by subdeterminants

Abstract

Keywords

ASJC Scopus subject areas

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