Equilibrium distributions of electrons on smooth plane conductors

Korevaar, J.; Kortram, R.A.

doi:10.1016/1385-7258(83)90057-4

Cited by 7 publications

(3 citation statements)

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“…In the case s = 0 sharp estimates of the discrepancy between the equilibrium measure and the normalized counting measure supported on minimal energy configurations were obtained in [23,33] on simple closed curves of smoothness C 3,ǫ on the plane. We obtain the following result.…”

Section: S V Borodachovmentioning

confidence: 99%

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Borodachov¹

2012

Can. j. math.

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We consider the problem of minimizing the energy of N points repelling each other on curves in ℝ ∞d with the potential |x — y|—s, s ≥ 1, where |・| is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for s ≥ 2, the minimal pairwise distance in optimal configurations asymptotically equals L/N, N → 1, where L is the length of the curve.

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Section: S V Borodachovmentioning

confidence: 99%

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Borodachov¹

2012

Can. j. math.

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“…For very smooth Γ, the approximations show that ω(E) − ω N (E) = O(1/N ), uniformly for the subarcs E ⊂ Γ. Via a Stieltjes integral for the potential difference U ω − U ωN this estimate implies that E ω − E ωN is at most of order 1/N at distance ≥ > 0 from the curve, see Korevaar and coauthors Geveci and Kortram [18,22,23]. These papers also show that the order O(1/N ) is sharp except when Γ is a circle.…”

Section: The Problemmentioning

confidence: 99%

“…For a conductor K given by an analytic or C 3,α Jordan curve Γ in the plane, Pommerenke [31] and Korevaar and Kortram [18,23] have obtained close approximations to the Fekete points. For very smooth Γ, the approximations show that ω(E) − ω N (E) = O(1/N ), uniformly for the subarcs E ⊂ Γ.…”

Section: The Problemmentioning

confidence: 99%

Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Korevaar

Monterie

1998

Trans. Amer. Math. Soc.

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Abstract. Let ω denote the classical equilibrium distribution (of total charge 1) on a convex or C 1,α -smooth conductor K in R q with nonempty interior. Also, let ω N be any N th order "Fekete equilibrium distribution" on K, defined by N point charges 1/N at N th order "Fekete points". (By definition such a distribution minimizes the energy for N -tuples of point charges 1/N on K.) We measure the approximation to ω by ω N for N → ∞ by estimating the differences in potentials and fields,both inside and outside the conductor K. For dimension q ≥ 3 we obtain uniform estimates O(1/N 1/(q−1) ) at distance ≥ ε > 0 from the outer boundary Σ of K. Observe that E ω = 0 throughout the interior Ω of Σ (Faraday cage phenomenon of electrostatics), hence E ω N = O(1/N 1/(q−1) ) on the compact subsets of Ω.For the exterior Ω ∞ of Σ the precise results are obtained by comparison of potentials and energies. Admissible sets K have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries Σ and bounds on normal derivatives of interior and exterior Green functions.Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions µ N of equal point charges on Σ. In that context there is an important open problem for the sphere which is discussed at the end of the paper.

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