The Jacobian determinant revisited

Brezis, Haı̈m; Nguyên, Hoài-Minh

doi:10.1007/s00222-010-0300-9

Cited by 55 publications

(100 citation statements)

References 32 publications

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“…We just discuss here a few results in [33], reported by Mironescu in [63]. We finally mention that the Jacobian determinant was extensively studied in the literature; see, e.g., [4,5,28,38,39,44,51,52,64,65,66,67,80,81,85,86] and the references therein.…”

Section: This Impliesmentioning

confidence: 90%

“…As a natural continuation of the study of the Jacobian distributional discussed in Section 6, with Brezis in [33], we deal with the Jacobian determinant. One of our goals is devoted to the search of an "optimal " space (containing all the above cases) in which the Jacobian determinant is well defined (note, for example, that neither W 1, a subset of the other).…”

Section: The Jacobian Determinantmentioning

confidence: 99%

“…For this purpose it is convenient to work in the fractional Sobolev spaces W s,p (Ω). We prove the following result [33,Theorem 3].…”

Section: The Jacobian Determinantmentioning

confidence: 99%

“…These results are obtained in collaboration with Bourgain and Brezis in [18], Bourgain in [21], and Brezis in [32,33,34], and by the author in [68,69,70,71,72,73,74]. The first topic is on estimates for the topological degree of maps from a sphere into itself.…”

Section: Introductionmentioning

confidence: 96%

“…More precisely, we discuss variants of the Sobolev inequality, the Poincaré inequality, and the Rellich-Kondrachov compactness criterion in these settings. The next two topics, joint work with Brezis in [32,33], are on the Jacobian distributional of maps from a sphere into itself and the Jacobian determinant. These are presented in Sections 6 and 7.…”

Section: Introductionmentioning

confidence: 99%

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Estimates for the topological degree and related topics

Nguyên

2014

J. Fixed Point Theory Appl.

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Abstract. This is a survey paper on estimates for the topological degree and related topics which range from the characterizations of Sobolev spaces and BV functions to the Jacobian determinant and nonlocal filter problems in Image Processing. These results are obtained jointly with Bourgain and Brezis. Several open questions are mentioned.Mathematics Subject Classification. 55M25, 49J99, 46E35, 46B50.

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Section: This Impliesmentioning

confidence: 90%

Section: The Jacobian Determinantmentioning

confidence: 99%

“…For this purpose it is convenient to work in the fractional Sobolev spaces W s,p (Ω). We prove the following result [33,Theorem 3].…”

Section: The Jacobian Determinantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Estimates for the topological degree and related topics

Nguyên

2014

J. Fixed Point Theory Appl.

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Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

Henao

Serfaty

2012

Comm Pure Appl Math

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We consider the minimization of $\int_{\Omega _\varepsilon } {|D{\bf u}|^p } d{\bf x}$ in a perforated domain $\Omega _\varepsilon : = \Omega \backslash \cup _{i = 1}^M B_\varepsilon ({\bf a}_i )$ of $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^n$ among maps $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf u} \in W^{1,p} (\Omega _\varepsilon ,\R^n )$ that are incompressible (det $D{\bf u} \equiv 1$) and invertible, and satisfy a Dirichlet boundary condition u = g on ∂Ω. If the volume enclosed by g(∂Ω) is greater than |Ω|, any such deformation u is forced to map the small holes Bε(ai) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of $\sum\nolimits_{i = 1}^M {v_i |\log \varepsilon |}$ and the model is suited, therefore, for an asymptotic analysis (v1,…, vM denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg‐Landau theory, we obtain estimates for the “renormalized” energy showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a1,…,aM, and on the distance from these points to the outer boundary ∂Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.

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Hölder-topology of the Heisenberg group

Schikorra

2020

Aequat. Math.

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The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions.We report on some recent progress in the Analysis of the Hölder topology of the Heisenberg group, some related and some unrelated to density questions for Sobolev maps into the Heisenberg group.In particular we describe the main ideas behind a result by Haj lasz, Mirra, and the author regarding Gromov's conjecture, which is based on the linking number. We do not prove or disprove the Gromov Conjecture.

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The Jacobian determinant revisited

Cited by 55 publications

References 32 publications

Estimates for the topological degree and related topics

Estimates for the topological degree and related topics

Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

Hölder-topology of the Heisenberg group

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