Abstract
In this paper, we investigate Riesz energy problems on unbounded conductors in \(\mathbb {R}^d\) in the presence of general external fields Q, not necessarily satisfying the growth condition \(Q(x)\rightarrow +\infty \) as \(|x|\rightarrow +\infty \) assumed in several previous studies. We provide sufficient conditions on Q for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor \(\mathbb {R}^{d}\), embedded in \(\mathbb {R}^{d+1}\), when the external field is created by the potential of a signed measure \(\nu \) supported outside of \(\mathbb {R}^{d}\). Simple cases where \(\nu \) is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vallée Poussin, are established. These results are of independent interest.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Alternative definition of a capacity of a compact set K due to de La Vallée Poussin is given as \(\textrm{cap}(K)=\max \mu (K)\), where the maximum is taken over positive measures supported on K such that \(U^\mu (x)\le 1\) on \(S_\mu \), the support of \(\mu \) (see [26, p. 139]).
To avoid ambiguity, we denote by \(\infty \) the Alexandroff point of \({{\mathbb {R}}}^d\) and use the explicit notations of \(-\infty \) and \(+\infty \) in the context of the real line.
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, Illinois (1970)
Bartle, R.G.: A Modern Theory of Integration. Graduate Studies in Mathematics, 32. American Mathematical Society, Providence, RI (2001)
Benko, D., Dragnev, P., Orive, R.: On point-mass Riesz external fields on the real axis. J. Math. Anal. Appl. 491, 124299 (2020)
Benko, D., Dragnev, P.D., Totik, V.: Convexity of harmonic densities. Rev. Mat. Iberoam. 28(4), 947–960 (2012)
Bilogliadov, M.: Minimum Riesz energy problem on the hyperdisk. J. Math. Phys. 59, 013301 (2018)
Bloom, T., Levenberg, N., Wielonsky, F.: Logarithmic potential theory and large deviation. Comput. Meth. Funct. Th. 15(4), 555–594 (2015)
Bloom, T., Levenberg, N., Wielonsky, F.: A large deviation principle for weighted Riesz interactions. Constr. Approx. 47, 119–140 (2018)
Borodachov, S.V., Hardin, D.P., Saff, E.B.: Discrete Energy on Rectifiable Sets. Springer Monographs in Mathematics, Springer, New York (2019)
Brauchart, J.S., Dragnev, P.D., Saff, E.B.: Riesz extremal measures on the sphere for axis-supported external fields. J. Math. Anal. Appl. 356, 769–792 (2009)
Brelot, M.: Sur le rôle du point à l’infini dans la théorie des fonctions harmoniques. Ann. Sci. Ecole Norm. Sup. 61, 301–332 (1944)
Brelot, M.: Lectures in Potential Theory. Tata Institute, Bombay (1960)
Bucur, C.: Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15, 657–699 (2016)
Chafaï, D., Gozlan, N., Zitt, P.A.: First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371–2413 (2014)
de La Vallée-Poussin, Ch.: Potentiel et problème généralisé de Dirichlet. Math. Gazette Lond. 22, 17–36 (1938)
Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin (1984)
Dragnev, P.D., Saff, E.B.: Riesz spherical potentials with external fields and minimal energy points separation. Potent. Anal. 26, 139–162 (2007)
Fuglede, B.: On the theory of potentials in locally compact spaces. Acta Math. 103(3–4), 139–215 (1960)
Fuglede, B.: Some properties of the Riesz charge associated with a \(\delta \)-subharmonic function. Potent. Anal. 1, 355–371 (1992)
Fuglede, B., Zorii, N.: Green kernels associated with Riesz kernels. Ann. Acad. Sci. Fenn. Math. 43, 121–145 (2018)
Hardy, A.: A note on large deviations for 2D Coulomb gas with weakly confining potential. Electron. Commun. Probab. 17(19), 1–12 (2012)
Hardy, A., Kuijlaars, A.B.J.: Weakly admissible vector equilibrium problems. J. Approx. Th. 164(6), 854–868 (2012)
Janssen, K.: On the Choquet charge of \(\delta \)-superharmonic functions. Potent. Anal. 12, 211–220 (2000)
Kuijlaars, A.B.J., Dragnev, P.: Equilibrium problems associated with fast decreasing polynomials. Proc. Am. Math. Soc. 127, 1065–1074 (1999)
Kurokawa, T., Mizuta, Y.: On the order at infinity of Riesz potentials. Hiroshima Math. J. 9(2), 533–545 (1979)
Leblé, T., Serfaty, S.: Large deviation principle for empirical fields of log and Riesz gases. Invent. Math. 210(3), 645–757 (2017)
Landkof, N.S.: Foundations of Modern Potential Theory, Grundlehren der mathematischen Wissenschaften 180. Springer-Verlag, New York-Heidelberg (1972)
García, A. López: Greedy energy points with external fields. Contemp. Math. 507, 189–207 (2010)
Mizuta, Y.: Potential Theory in Euclidean Spaces. Gakuto International Series. Mathematical Sciences and Applications, 6. Gakkotosho Co., Ltd., Tokyo (1996)
Ohtsuka, M.: On potentials in locally compact spaces. J. Sci. Hiroshima Univ. Ser. A-I Math. 25, 135–352 (1961)
Orive, R., Lara, J. Sánchez, Wielonsky, F.: Equilibrium problems in weakly admissible external fields created by pointwise charges. J. Approx. Theory 244, 71–100 (2019)
Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)
Riesz, M.: Intégrales de Riemann-Liouville et potentiels. Acta Szeged 9, 1–42 (1938)
Riesz, M.: L’intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 81, 1–223 (1949)
Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften, vol. 316. Springer-Verlag, Berlin (1997)
Simeonov, P.: A weighted energy problem for a class of admissible weights. Houston J. Math. 31(4), 1245–1260 (2005)
Sodin, M.: Hahn decomposition for the Riesz charge of \(\delta \)-subharmonic functions. Math. Scand. 83(2), 277–282 (1998)
Zorii, N.: On the solvability of the Gauss variational problem. Comput. Meth. Funct. Th. 2, 427–448 (2002)
Zorii, N.V.: Equilibrium potentials with external fields. Ukrainian Math. J. 55(9), 1423–1444 (2003)
Zorii, N.: Equilibrium problems for infinite dimensional vector potentials with external fields. Potent. Anal. 38(2), 397–432 (2013)
Zorii, N.: Harmonic measure, equilibrium measure, and thinness at infinity in the theory of Riesz potentials. Potent. Anal. (2021). https://doi.org/10.1007/s11118-021-09923-2
Acknowledgements
We thank N.V. Zorii for useful comments on this paper. The research of the first author was supported in part by NSF grant DMS-1936543. The research of the second author was partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades, under grant MTM2015-71352-P, by Spanish Ministerio de Ciencia e Innovación, under grant PID2021-123367NB-100, and by the PFW Scholar-in-Residence program.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Wolfgang Dahmen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dragnev, P.D., Orive, R., Saff, E.B. et al. Riesz Energy Problems with External Fields and Related Theory. Constr Approx 57, 1–43 (2023). https://doi.org/10.1007/s00365-022-09588-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-022-09588-z