Locally one-to-one mappings and a classical theorem on schlicht functions

Meisters, Gary H.; Olech, Czesraw

doi:10.1215/s0012-7094-63-03008-4

Cited by 70 publications

(18 citation statements)

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“…Also remember that, by construction, ϕ is C k close enough to the identity in Ω 0 and is the identity on ∂Ω 0 so that ϕ(t) is a C k −diffeomorphism (see Appendix A.6 and the topological arguments of [41]).…”

Section: The Fixed Point Methodsmentioning

confidence: 99%

“…We would like to obtain that ϕ is in fact a global diffeomorphism from Ω 0 into itself. This extension needs topological arguments contained in the article [41] of Meisters and Olech.…”

Section: A6 Globalization Of Local Diffeomorphismsmentioning

confidence: 99%

“…Theorem 1 of [41] applied to the ball of R N writes as follows. In fact, the original result of [41] includes different domains than the balls.…”

Section: A6 Globalization Of Local Diffeomorphismsmentioning

confidence: 99%

“…Theorem 1 of [41] applied to the ball of R N writes as follows. In fact, the original result of [41] includes different domains than the balls. Nevertheless, if we consider any smooth domain, then it has to be diffeomorphic to a ball (typically, annulus are excluded).…”

Section: A6 Globalization Of Local Diffeomorphismsmentioning

confidence: 99%

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Schrödinger Equation in Moving Domains

Duca

Joly

2021

Ann. Henri Poincaré

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We consider the Schrödinger equationon Ω(t) ( * )where Ω(t) ⊂ R N is a moving domain depending on the time t ∈ [0, T ]. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain Ω0 and a specific family of unitary maps h (t) : L 2 (Ω(t), C) −→ L 2 (Ω0, C). We show that the conjugation by h provides a new equation of the formwhere h = (h ) −1 . The Hamiltonian H(t) is a magnetic Laplacian operator of the formwhere A is an explicit magnetic potential depending on the deformation of the domain Ω(t). The formulation ( * * ) enables to ensure the existence of weak and strong solutions of the initial problem ( * ) on Ω(t) endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for ( * ) are not the homogeneous but the magnetic ones ∂ν u(t) + i ν|A u(t) = 0, even though ( * ) has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and a time-dependent version of the socalled "Moser's trick". We use this outcome in order to simplify Equation ( * * ) and to guarantee the well-posedness for slightly less regular deformations of Ω(t).

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Section: The Fixed Point Methodsmentioning

confidence: 99%

“…We would like to obtain that ϕ is in fact a global diffeomorphism from Ω 0 into itself. This extension needs topological arguments contained in the article [41] of Meisters and Olech.…”

Section: A6 Globalization Of Local Diffeomorphismsmentioning

confidence: 99%

“…Theorem 1 of [41] applied to the ball of R N writes as follows. In fact, the original result of [41] includes different domains than the balls.…”

Section: A6 Globalization Of Local Diffeomorphismsmentioning

confidence: 99%

Section: A6 Globalization Of Local Diffeomorphismsmentioning

confidence: 99%

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Schrödinger Equation in Moving Domains

Duca

Joly

2021

Ann. Henri Poincaré

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“…It follows by a theorem of Meisters and Olech [10], that J is a diffeomorphism of E . For 3 sufficiently small so that E lies in the interior of +#,,…”

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confidence: 95%

The Graves Condition for Incompressible Finite Elasticity

Sithigh

2003

Mathematics and Mechanics of Solids

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7R 3URI 0LOODUG ) %HDWW\ RQ WKH RFFDVLRQ RI KLV UHWLUHPHQW$EVWUDFW The Graves and Weierstrass necessary conditions for minimizers are of central importance in the study of elastic materials that can undergo a change of phase. Proofs of these results applicable to the case of unconstrained materials abound. For the case of an incompressible material, however, only partial results have been established. Here, a complete proof is given. ,1752'8&7,21In 1975, Ericksenòs profoundly influential paper [1] on the equilibrium of bars appeared, and inaugurated a period of intensive and fruitful research into phase transformations in solids. In studies of this sort, the material response is generally modeled by finite elasticity. Phase boundaries are treated as surfaces of discontinuity across which the deformation gradient field experiences large jumps. Such a deformation is regarded as stable if it minimizes the associated energy functional.This research lent fresh relevance to the classical necessary conditions of the calculus of variations for weak and strong relative minimizers. In particular, the Weierstrass necessary condition proved to be of central importance. Its physical import in this particular context is that strong relative minimizers of energy must satisfy the Maxwell relation at any phase boundary they contain. Various proofs of this condition applicable to the case of unconstrained elasticity are to be found in the classical literature of the calculus of variations. A representative one is that of Graves [2]. The case of incompressible elasticity, i.e. of elastic materials that are constrained to undergo only deformations which are isochoric, is more difficult, and is the subject of the present study.The prior literature on the question is rather sparse. Ericksen [3], in a discussion of the Weierstrass condition, stated the form which he thought the analogous condition for the incompressible case ought to take. Abeyaratne [4] considered the equilibrium problem for coexistent phases in incompressible elastic materials without discussing stability. He later [5] concluded a very nice discussion of the Maxwell and related conditions for the unconstrained 0DWKHPDWLFV DQG 0HFKDQLFV RI 6ROLGV 289ï298, 2003

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Image Morphing Using Deformation Techniques

Lee

Chwa

Hahn

et al. 1996

J. Visual. Comput. Animat.

101

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This paper presents a new image morphing method using a two‐dimensional deformation technique which provides an intuitive model for a warp. The deformation technique derives aC1‐continuous and one‐to‐one warp from a set of point pairs overlaid on two images. The resulting in‐between image precisely reflects the correspondence of features specified by an animator. We also control the transition behaviour in a metamorphosis sequence by taking another deformable surface model, which is simpler and thus more efficient than the deformation technique for a warp. The proposed method separates transition control from feature interpolation and is easier to use than the previous techniques. The multigrid relaxation method is employed to solve a linear system in deriving a warp or transition rates. This method makes our image morphing technique fast enough for an interactive environment.

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Locally one-to-one mappings and a classical theorem on schlicht functions

Cited by 70 publications

References 0 publications

Schrödinger Equation in Moving Domains

Schrödinger Equation in Moving Domains

The Graves Condition for Incompressible Finite Elasticity

Image Morphing Using Deformation Techniques

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