Smoothed projections over weakly Lipschitz domains

Licht, Martin W.

doi:10.1090/mcom/3329

Cited by 7 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result is a consequence of [37, Theorem 7.4 & Corollary 7.5]; details can be found for instance in [36,Theorem 2.3]. The extension of u k is then achieved via reflection.…”

Section: Cartesian Currentsmentioning

confidence: 74%

The N-Clock Model: Variational Analysis for Fast and Slow Divergence Rates of N

Cicalese

Orlando

Ruf

2022

Arch Rational Mech Anal

View full text Add to dashboard Cite

We study a nearest neighbors ferromagnetic classical spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of N equi-spaced vectors, also known as the N-clock model. We find a fast rate of divergence of N with respect to the lattice spacing for which the N-clock model has the same discrete-to-continuum variational limit as the classical XY model (also known as planar rotator model), in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of N at which the coarse-grain limit does not detect topological defects, but it is instead a BV-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for N, whose analysis requires the aid of Cartesian currents.

show abstract

“…This result is a consequence of [37, Theorem 7.4 & Corollary 7.5]; details can be found for instance in [36,Theorem 2.3]. The extension of u k is then achieved via reflection.…”

Section: Cartesian Currentsmentioning

confidence: 74%

The N-Clock Model: Variational Analysis for Fast and Slow Divergence Rates of N

Cicalese

Orlando

Ruf

2022

Arch Rational Mech Anal

View full text Add to dashboard Cite

show abstract

“…There have been kinds of interpolators to finite element spaces which work for functions with minimal regularity requirements, such as [16,19,23,24,26,31,32,41]. For these interpolators, the regularization, smoothing or averaging techniques are usually used based on macroelements consisting of patches of elements.…”

Section: Andmentioning

confidence: 99%

Partially adjoint discretizations of adjoint operators

Zhang¹

2022

Preprint

View full text Add to dashboard Cite

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. The validities of the theoretical framework and the formal construction of discretizations are illustrated by a systematic family of in-pair discretizations of the adjoint exterior differential operators.The adjoint properties concerned in the paper are the closed range theorem and the strong dualities, whose preservations have not been well studied yet. Quantified versions of the closed range theorem are established for both adjoint operators and partially adjoint discretizations. The notion Poincaré-Alexander-Lefschetz (P-A-L for short) type duality is borrowed for operator theory, and horizontal and vertical P-A-L dualities are figured out for adjoint operators and their analogues are established for partially adjoint discretizations. Particularly by partially adjoint discretizations of exterior differential operators, the Poincaré-Lefschetz duality is preserved as an identity, which was not yet obtained before. The ABCD, a new kind of discretization method, is motivated by and plays a crucial role in the construction of partially adjoint discretizations. Besides, it can be used for the discretization of single operators; for example, in this paper, de Rham complexes that start with the Crouzeix-Raviart finite element spaces are constructed by series of ABCDs, including both the discretized operators and the domain spaces; commutative diagrams with appropriate regularities are constructed thereon. Equivalences are established between primal and dual discretizations for the elliptic source and eigenvalue problems and the primal and dual mixed discretizations of the Hodge Laplacian problems of exterior differential operators based on the partially adjoint discretizations of adjoint operators. This shows how structure-preserving schemes for equations with linear operators can be lead to by partially adjoint discretizations under the new theoretical framework. SHUO ZHANG 1.3. Organization of the paper 7 1.4. Notations and conventions 8 2. Theory of partially adjoint operators 8 2.1. Basics of adjoint operators revisited 8 2.2. Theory of partially adjoint operators 10 2.3. Some technical proofs 19 3. Partially adjoint discretizations of adjoint operators 23 3.1. Partially adjoint discretizations by CD and ABCD 24 3.2. Characteristics of lo...

show abstract

“…We start by finding a family of open, bounded, weakly Lipschitz, and simply connected sets (Ω s ) s compactly contained in Ω that exhausts Ω . To do so, we rely on a result proven in [29,Theorem 2.3] (see also [30,Theorem 7.4 and Corollary 7.5]), which guarantees that there exists a t > 0 and a bi-Lipschitz map Γ : ∂Ω×(−t, t) → Γ(∂Ω×(−t, t)) ⊂ R 2 such that the image of Γ is an open neighborhood of ∂Ω , Γ(x, 0) = x for x ∈ ∂Ω , Γ(∂Ω×(−t, 0)) ⊂ Ω , and Γ(∂Ω×(0, t)) ⊂ R 2 \ Ω . For s ∈ (0, t) we define the set Ω s = Ω \ Γ(∂Ω×[−s, 0)) ⊂ Ω .…”

Section: From (34) It Follows Thatmentioning

confidence: 99%

“…For s ∈ (0, t) we define the set Ω s = Ω \ Γ(∂Ω×[−s, 0)) ⊂ Ω . According to [29,Theorem 2.3], we can assume that each Ω s is an open, bounded, and weakly Lipschitz set. We claim that each Ω s is simply connected.…”

Section: From (34) It Follows Thatmentioning

confidence: 99%

Variational Analysis of a Two-Dimensional Frustrated Spin System: Emergence and Rigidity of Chirality Transitions

Cicalese¹,

Forster²,

Orlando³

2019

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We study the discrete-to-continuum variational limit of the J 1 -J 3 spin model on the square lattice in the vicinity of the helimagnet/ferromagnet transition point as the lattice spacing vanishes. Carrying out the Γ -convergence analysis of proper scalings of the energy, we prove the emergence and characterize the geometric rigidity of the chirality phase transitions.where J 1 and J 3 are positive constants (the interaction parameters of the model) and for every lattice point σ ∈ Z 2 we denote by u σ the value of the spin variable u at σ . The first term in the energy is ferromagnetic as it favors aligned nearest-neighboring spins, whereas the second one is antiferromagnetic as it favors antipodal third-neighboring spins. In the case where J 3 = 0 the energy describes the so-called XY model, whose variational analysis has been carried out in [2,9,18] also in connection with the theory of dislocations [4,5,6]. When J 3 > 0 , the ferromagnetic and the antiferromagnetic terms in E compete. For an 1 arXiv:1904.07792v1 [math.AP]

show abstract

Smoothed projections over weakly Lipschitz domains

Cited by 7 publications

References 29 publications

The N-Clock Model: Variational Analysis for Fast and Slow Divergence Rates of N

The N-Clock Model: Variational Analysis for Fast and Slow Divergence Rates of N

Partially adjoint discretizations of adjoint operators

Variational Analysis of a Two-Dimensional Frustrated Spin System: Emergence and Rigidity of Chirality Transitions

Contact Info

Product

Resources

About