scholarly journals Nonlocal-interaction equations on uniformly prox-regular sets

2015 ◽  
Vol 36 (3) ◽  
pp. 1209-1247
Author(s):  
Lijiang Wu ◽  
Dejan Slepčev ◽  
José A. Carrillo
Keyword(s):  
Nonlocal Interaction ◽  
Regular Sets

scholarly journals The nonlocal-interaction equation near attracting manifolds

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Francesco S. Patacchini ◽  
Dejan Slepčev
Keyword(s):  
Numerical Experiments ◽  
Compact Manifold ◽  
Attractive Potential ◽  
Nonlocal Interaction ◽  
Well Posedness ◽  
Gradient Flows ◽  
Compact Sets ◽  
Interaction Equation ◽  
Sets With Positive Reach ◽  
Regular Sets

<p style='text-indent:20px;'>We study the approximation of the nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> embedded in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula>, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> can be approximated by the classical nonlocal-interaction equation on <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> by adding an external potential which strongly attracts to <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. The proof relies on the Sandier–Serfaty approach [<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b24">24</xref>] to the <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on <inline-formula><tex-math id="M7">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, which was shown [<xref ref-type="bibr" rid="b10">10</xref>]. We also provide an another approximation to the interaction equation on <inline-formula><tex-math id="M8">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, based on iterating approximately solving an interaction equation on <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> and projecting to <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.</p>

scholarly journals Prox-regular sets and Legendre-Fenchel transform related to separation properties

Optimization ◽  
2020 ◽  
pp. 1-33
Author(s):  
Samir Adly ◽  
Florent Nacry ◽  
Lionel Thibault
Keyword(s):  
Fenchel Transform ◽  
Regular Sets

scholarly journals Plenty of big projections imply big pieces of Lipschitz graphs

2021 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Graphs ◽  
Regular Sets ◽  
Big Pieces

AbstractI prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

Buzz and pipelines: the costs and benefits of local and nonlocal interaction

2018 ◽  
Vol 19 (3) ◽  
pp. 753-773 ◽  
Author(s):  
Christopher R Esposito ◽  
David L Rigby
Keyword(s):  
Nonlocal Interaction ◽  
Costs And Benefits

Equilibrium measure for a nonlocal dislocation energy with physical confinement

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria Giovanna Mora ◽  
Alessandro Scagliotti
Keyword(s):  
Equilibrium Measure ◽  
Symmetric Case ◽  
Nonlocal Interaction ◽  
Interaction Energies ◽  
Coulomb Interactions ◽  
Positive Edge ◽  
Dislocation Energy ◽  
Explicit Measure ◽  
Edge Dislocations

Abstract In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I α I_{\alpha} that describe the interaction of particles confined in an elliptic subset of the plane. The case α = 0 \alpha=0 corresponds to purely Coulomb interactions, while the case α = 1 \alpha=1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter 𝛼 and favors the alignment of particles. We show that the equilibrium measure is completely unaffected by the anisotropy and always coincides with the optimal distribution in the case α = 0 \alpha=0 of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result does not seem to agree with the mechanical conjecture that positive edge dislocations at equilibrium tend to arrange themselves along “wall-like” structures. Moreover, this is one of the very few examples of explicit characterization of the equilibrium measure for nonlocal interaction energies outside the radially symmetric case.

Equational axioms for regular sets

1993 ◽  
Vol 3 (1) ◽  
pp. 1-24 ◽  
Author(s):  
S. L. Bloom ◽  
Z. Ésik
Keyword(s):  
Star Operation ◽  
Regular Sets

We show that, aside from the semiring equations, three equations and two equation schemes characterize the semiring of regular sets with the Kleene star operation.

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