arXiv subjects

Mykhailo Bilogliadov

Publications and source records attributed to Mykhailo Bilogliadov.

4 recordsLinked to original sources

arXiv2016

Minimum Riesz Energy Problem on the Hyperdisk

We consider the minimum Riesz $s$-energy problem on the unit disk $\mathbb D:=\{(x_1,\ldots,x_d)\in\mathbb R^d: x_1=0, x_2^2+x_3^2+\ldots+x_d^2\leq 1\}$ in the Euclidean space $\mathbb R^d$, $d\geq 3$, immersed into a smooth rotationally invariant external field $Q$. The charges are assumed to interact via the Riesz potential $1/r^s$, with $d-3 < s < d-1$, where $r$ denotes the Euclidean distance. We solve the problem by finding an explicit expression for the extremal measure. We then consider applications to a monomial external field and an external field generated by a positive point charge, located at some distance above the disk on the polar axis. We obtain an equation describing the critical height for the location of the point charge, which guarantees that the support of the extremal measure occupies the whole disk $\mathbb D$. We also show that under some mild restrictions on a general external field the support of the extremal measure will have a ring structure. Furthermore, we demonstrate how to reduce the problem of recovery of the extremal measure in this case to a Fredholm integral equation of the second kind.

math.CA↗

arXiv2016

Minimum Energy Problem on the Hypersphere

We consider the minimum energy problem on the unit sphere $\mathbb S^{d-1}$ in the Euclidean space $\mathbb R^d$, $d\geq 3$, in the presence of an external field $Q$, where the charges are assumed to interact according to Newtonian potential $1/r^{d-2}$, with $r$ denoting the Euclidean distance. We solve the problem by finding the support of the extremal measure, and obtaining an explicit expression for the density of the extremal measure. We then apply our results to an external field generated by a point charge of positive magnitude, placed at the North Pole of the sphere, and to a quadratic external field.

math.CA↗

arXiv2015

Weighted energy problem on the unit sphere

We consider the minimal energy problem on the unit sphere $\mathbb S^2$ in the Euclidean space $\mathbb R^3$ immersed in an external field $Q$, where the charges are assumed to interact via Newtonian potential $1/r$, $r$ being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to the external field generated by a point charge, and a quadratic external field.

math.CA↗

arXiv2014

Equilibria of the field generated by point charges

We consider a special case of Maxwell's problem on the number of equilibrium points of the Riesz potential $1/r^{2β}$ (where $r$ is the Euclidean distance and $β$ is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges $n$ and the Riesz parameter $β$. Finally, we prove that for values of $β$ in a small neighborhood of $β=1$ the Riesz potential has only one equilibrium point different from the origin on each perpendicular bisector, and one equilibrium point at the origin.

math.CA↗