Area Formulas for σ-Harmonic Mappings

Alessandrini, Giovanni; Nesi, Vincenzo

doi:10.1007/978-1-4615-0777-2_1

Cited by 4 publications

(7 citation statements)

References 47 publications

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“…The final Section 6 collects further developments, remarks and connections with various relevant areas and applications. In § 6.1 we extend some area formulas first discussed in [9]. In § 6.2 we lay a bridge towards the theory of correctors in homogenization.…”

Section: σ-Harmonic Mappingsmentioning

confidence: 93%

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Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds

Alessandrini¹,

Nesi²

2007

Preprint

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In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called periodic σ-harmonic sense preserving homeomorphisms of C onto itself.

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Section: σ-Harmonic Mappingsmentioning

confidence: 93%

“…In § 6.2 we lay a bridge towards the theory of correctors in homogenization. Finally § 6.3 develops an application of the Theorem by Astala [11], generalizing results in [33] and [9].…”

Section: σ-Harmonic Mappingsmentioning

confidence: 99%

Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds

Alessandrini¹,

Nesi²

2007

Preprint

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“…where Hess( z ) is the Hessian of z . This operator is particularly relevant in the computation of quasi‐harmonic fields [AN02] and in elasticity [Hu54]. Graphics applications have also used this generalized Laplacian to compute anisotropic parameterization [ZRS05, KMZ11] and filtering [PM90], and more recently to design simplicial masonry structures [dGAOD13, LHS*13].…”

Section: Tensor Fields Over Smooth Surfacesmentioning

confidence: 99%

“…As mentioned in §2.4, the Laplacian operator of functions, commonly used in geometry processing, is a particular case of a general family of differential operators on functions Δ σ ( z ) = div (σ∇ z ) where σ is a symmetric 2‐tensor field [AN02]. We can define its weak (integrated) form on a discrete scalar function z = Σ j z j φ j as…”

Section: Discrete Differential Tensor‐based Operatorsmentioning

confidence: 99%

“…(Dual 2‐forms, defined per dual cell, can also be used.) In comparison, symmetric tensors have rarely been discussed in the discrete realm [Mei03, KASH13], yet they are implicitly behind all inner products of forms or vectors, generalized Laplacian operators [AN02], and the notion of Hodge star [BH06]. Our discrete encoding of tensors encompasses both symmetric and antisymmetric tensors in a consistent framework.…”

Section: Introductionmentioning

confidence: 99%

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Discrete 2‐Tensor Fields on Triangulations

Goes

Liu²,

Budninskiy

et al. 2014

Computer Graphics Forum

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Geometry processing has made ample use of discrete representations of tangent vector fields and antisymmetric tensors (i.e., forms) on triangulations. Symmetric 2-tensors, while crucial in the definition of inner products and elliptic operators, have received only limited attention. They are often discretized by first defining a coordinate system per vertex, edge or face, then storing their components in this frame field. In this paper, we introduce a representation of arbitrary 2-tensor fields on triangle meshes. We leverage a coordinate-free decomposition of continuous 2-tensors in the plane to construct a finite-dimensional encoding of tensor fields through scalar values on oriented simplices of a manifold triangulation. We also provide closed-form expressions of pairing, inner product, and trace for this discrete representation of tensor fields, and formulate a discrete covariant derivative and a discrete Lie bracket. Our approach extends discrete/finite-element exterior calculus, recovers familiar operators such as the weighted Laplacian operator, and defines discrete notions of divergence-free, curl-free, and traceless tensors-thus offering a numerical framework for discrete tensor calculus on triangulations. We finally demonstrate the robustness and accuracy of our operators on analytical examples, before applying them to the computation of anisotropic geodesic distances on discrete surfaces.

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