A minimum energy problem and Dirichlet spaces

Abstract

Abstract. We analyze a minimum energy problem for a discrete electrostatic model in the complex plane and discuss some applications. A natural characteristic distinguishing the state of minimum energy from other equilibrium states is established. It enables us to gain insight into the structure of positive trigonometric polynomials and Dirichlet spaces associated with finitely atomic measures. We also derive a related family of linear second order differential equations with polynomial solutions.

The Growth of Healthy and Cancerous Tissues

“…The above forming symmetry well explained by the Dirichlet theorem of minimizing the electrostatic energy, [15]. Consider the cells conductive, having resting potential (70 -100 mV).…”

Electrostatic Equilibria on the Unit Circle via Jacobi Polynomials

“…Recall the convention that if a particle of charge q is located at a point a ∈ C and a particle of charge p is located at a point b ∈ C, then the force on the particle at b due to the particle at a is 2pq/( b − ā) (as in [2,3,7]). With this convention, we have the following lemma relating critical points of general Hamiltonians of the form (2) to the condition of electrostatic equilibrium (see also [2]). Lemma 3.…”

“…as in [1,2], where the particles at the points {e it j } M j=1 are considered "mobile," the particles at {e iη j } K j=1 are considered "fixed," and σ(x) > 0 denotes the charge carried by the particle located at x ∈ C. To avoid any ambiguity that may arise from rotating the circle, we will always assume K ≥ 1.…”

An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

“…Our main application of the results in Section 1 will be to demonstrate that the zeros of certain families of POPUC are the locations of points that are in electrostatic equilibrium. We will use the convention (as in [11]) that if a particle of charge q is located at a point a ∈ C and a particle of charge p is located at a point b ∈ C, then the force on the particle at b due to the particle at a is 2pq/( b − ā).…”

“…. , n (see [11,22]). Similarly, the condition ( 6) is satisfied if and only if the force exerted on the particle at x j is equal to zero for each j = 1, .…”