Geometrical logarithmic capacitance

Xiao, Jie

doi:10.1016/j.aim.2020.107048

Cited by 10 publications

(3 citation statements)

References 52 publications

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“…Scheven [8]. Inequalities for the p-capacity were proved very recently by J. Xiao [44] and E. Mukoseeva [30].…”

Section: 2)mentioning

confidence: 93%

Condenser capacity and hyperbolic perimeter

Nasser,

Rainio,

Vuorinen

2021

Preprint

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We apply domain functionals to study the conformal capacity of condensers (G, E) where G is a simply connected domain in the complex plane and E is a compact subset of G. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of E. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.

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“…Scheven [8]. Inequalities for the p-capacity were proved very recently by J. Xiao [44] and E. Mukoseeva [30].…”

Section: 2)mentioning

confidence: 93%

Condenser capacity and hyperbolic perimeter

Nasser,

Rainio,

Vuorinen

2021

Preprint

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“…Given this result, the uniqueness of logarithmic equilibrium measure follows quickly as in the planar case, by considering two equilibrium measures and taking their average to deduce that the measures must in fact agree. Incidentally, many geometric properties of logarithmic capacity in higher dimensions have been investigated recently by Xiao [32,33].…”

Section: Potential Theoretic Backgroundmentioning

confidence: 99%

Minimizing capacity among linear images of rotationally invariant conductors

Laugesen¹

2021

Preprint

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Logarithmic capacity is shown to be minimal for a planar set having N -fold rotational symmetry (N ≥ 3), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is Pólya and Schiffer's lower bound on capacity in terms of moment of inertia.

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“…In this case, if H is the Euclidean norm (i.e. H(ξ) = |ξ|), the model (1.2)- (1.3) applies to the study of logarithmic capacity [14,53], and it determines the N -equilibrium potential of Ω, which naturally appears in computing the capacitance difference between coaxial cylindrical capacitors (see [39]). Analogously, for a general H, problem (1.2)-(1.3) can be applied to the study of the related capacity problems, when the set Ω is embedded in a possibly anisotropic medium.…”

Section: Introductionmentioning

confidence: 99%

An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

Ciraolo¹,

Li²

2021

Preprint

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We consider a partially overdetermined problem for anisotropic N -Laplace equations in a convex cone Σ intersected with the exterior of a bounded domain Ω in R N , N ≥ 2. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that Σ ∩ Ω must be the intersection of the Wulff shape and Σ. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.

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Geometrical logarithmic capacitance

Cited by 10 publications

References 52 publications

Condenser capacity and hyperbolic perimeter

Condenser capacity and hyperbolic perimeter

Minimizing capacity among linear images of rotationally invariant conductors

An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

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