Abstract
This chapter deals with the continuous minimal energy problem on a compact set A in \(\mathbb R^p\) with respect to a symmetric and lower semicontinuous kernel K. The asymptotic behavior as N gets large of the discrete minimal energy problem on A (introduced in Section 2.1) is also studied when A has nonzero K-capacity; i.e., finite Wiener constant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
More precisely, we apply Fatou’s lemma to \(K(x_n, y)-c\), where c is a lower bound for K on \(A\times A\).
- 2.
The term minimizing measure is also used.
- 3.
Since K is lower semicontinuous, it is bounded below and the integral on the right-hand side is well defined (cf. Section 1.4).
- 4.
More generally, the conclusion holds for any Borel set B with all its compact subsets having K-capacity 0.
- 5.
At least one K-equilibrium measure exists on A in view of Lemma 4.1.3.
- 6.
A point x is called an atom for a positive measure \(\gamma \) if \(\gamma (\{x\})>0\).
- 7.
\(\mathcal D_m\) is the family of the cubes of the form \(\{(x_1,\ldots , x_p)\in \mathbb R^p : k_i2^{-m}\le x_i<(k_i+1)2^{-m},\ i=1,\ldots , p\}\), where \(k_1,\ldots , k_p\) are arbitrary integers.
- 8.
That \(h_n(z)\) is harmonic at \(\infty \) (or, equivalently, \(h_n(1/z)\) is harmonic at \(z=0\)) is a consequence of the fact that \(\lim \limits _{z\rightarrow \infty }h_n(z)\) exists and equals a finite constant (namely \(\log 2\)) so that \(h_n(z)\) has a removable singularity at \(\infty \).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
Borodachov, S.V., Hardin, D.P., Saff, E.B. (2019). Continuous Energy and Its Relation to Discrete Energy. In: Discrete Energy on Rectifiable Sets. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84808-2_4
Download citation
DOI: https://doi.org/10.1007/978-0-387-84808-2_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-84807-5
Online ISBN: 978-0-387-84808-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)