Potential theory

Potential theory is a branch of mathematical analysis that deals with potentials and potential functions, typically in relation to fields such as electrostatics, gravitation, fluid dynamics, and various areas of applied mathematics. The theory is largely concerned with the behavior of harmonic functions and their properties. At its core, potential theory examines the concept of a potential function, which describes gravitational or electrostatic potentials in physics.

Boundary value problems (BVPs) are a type of differential equation problem where one seeks solutions that satisfy specified conditions, or "boundary conditions," at certain values of the independent variable. These problems are prevalent in various fields of science and engineering, where they often arise in the context of physical systems described by differential equations.

Harmonic functions are a special class of functions that arise in various fields, including mathematics, physics, and engineering.

Subharmonic functions are a class of functions that arise in the study of potential theory and various fields of mathematics, including complex analysis and partial differential equations.

"An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism" is a significant work by the British mathematician and physicist George Gabriel Stokes, published in 1850. In this essay, Stokes explores the mathematical foundations and principles underlying the theories of electricity and magnetism, providing insights that bridge the gap between mathematical analysis and physical phenomena.

Axial multipole moments are a set of mathematical quantities used in physics, particularly in the study of electric and magnetic fields generated by charge and current distributions, respectively. They extend the concept of multipole expansions, which represent how a distribution of charges or currents influences the field at large distances from the source. 1. **Multipole Moments**: In general, multipole moments classify the behavior of the electric or magnetic field generated by a distribution as a function of distance from the source.

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Bessel potentials are a type of potential operator associated with Bessel functions, which are solutions to Bessel's differential equation. In functional analysis and partial differential equations, Bessel potentials are used to define certain types of Sobolev spaces and are closely related to the notion of fractional derivatives. The Bessel potential of order \( \alpha \) can be defined in terms of the Bessel operator.

Boggio's formula is a mathematical result used in the context of potential theory and solutions of the Poisson equation related to electrostatics. It provides a way to compute the potential (or electric field) due to point charges or other distributions under certain conditions. While there are various contexts in which the name "Boggio's formula" might arise, it is most commonly associated with the problem of determining the potential due to a point charge outside a sphere.

In mathematics, particularly in the fields of measure theory and set theory, the term "capacity" can refer to a few different concepts, depending on the context. Here's a brief overview: 1. **Set Capacity in Measure Theory**: In the context of measure theory, capacity is a way to generalize the concept of "size" of a set. The capacity of a set can refer to various types of measures assigned to sets that may not be measurable in the traditional sense.

Cylindrical multipole moments are a mathematical representation used in physics and engineering to describe the distribution of mass, charge, or any other physical quantity in a cylindrical coordinate system. These moments help in analyzing systems with cylindrical symmetry, such as wires, cylinders, or other structures that exhibit similar symmetry properties. ### Definition and Calculation Cylindrical multipole moments extend the concept of multipole moments, which are generally used to describe the spatial distribution of charges or masses in Cartesian coordinates.

A dipole generally refers to a system that has two equal but opposite charges or magnetic poles separated by a distance. There are two main contexts in which the term "dipole" is commonly used: 1. **Electric Dipole**: In electrostatics, an electric dipole consists of two equal and opposite electric charges (positive and negative) separated by a distance.

The Dirichlet problem is a type of boundary value problem that arises in mathematical analysis, particularly in the study of partial differential equations (PDEs). It involves finding a function that satisfies a certain differential equation within a domain, subject to specified values on the boundary of that domain.

The double layer potential is a concept from potential theory and is particularly relevant in the study of boundary value problems in mathematical physics, especially in the context of electrostatics and fluid dynamics. It is often used when dealing with boundary integral equations. ### Definition In a simple sense, the double layer potential is a way to represent a distribution of surface charges on a boundary in an n-dimensional space.

Ewald summation is a mathematical technique used to compute the potential energy and forces in systems with periodic boundary conditions, commonly encountered in simulations of charged systems or dipolar systems in condensed matter physics, materials science, and molecular dynamics. The main challenge in these systems is that the Coulomb potential between charges, which falls off as \(1/r\), leads to divergent sums when calculated directly for an infinite periodic lattice.

Extremal length is a concept from the field of complex analysis and geometric topology, specifically concerning the study of Riemann surfaces and conformal mappings. It is used to measure the size of families of curves on a surface and has applications in various areas, including Teichmüller theory and the study of conformal structures. Mathematically, the extremal length of a family of curves is defined via a certain optimization problem.

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The Furstenberg boundary is a concept in probability theory and dynamical systems, particularly in the study of random walks on groups and homogeneous spaces. Named after the mathematician Herbert Furstenberg, this boundary provides a way to understand the asymptotic behavior of random walks by relating them to geometric structures. In more detail, the Furstenberg boundary can be defined in the context of a probability measure on a group, often a non-abelian group.

Harmonic measure is a concept in mathematical analysis, particularly in potential theory and complex analysis. It is associated with harmonic functions, which are functions that satisfy Laplace's equation. Here are some key points to understand harmonic measure: 1. **Harmonic Functions**: A function \( u \) is harmonic in a domain if it is twice continuously differentiable and satisfies Laplace's equation, i.e., \( \nabla^2 u = 0 \).

Kellogg's theorem, in the context of topology and mathematical analysis, specifically deals with the behavior of continuous functions and the structure of spaces in relation to certain properties of sets. The theorem asserts that if a sequence of open sets in a topological space has certain convergence properties, then their limit behaves in a controlled manner.

Laplace expansion, also known as the Laplace transform, is a mathematical technique used to transform a function of time (often a signal or a system's response) into a function of a complex variable. The Laplace transform is especially useful in engineering and physics for analyzing linear time-invariant systems, particularly in control theory and circuit analysis.

The Lebesgue spine is a concept from measure theory, specifically in the context of Lebesgue integration and the study of measurable sets and functions. It refers to a specific construction related to the decomposition of measurable sets. More precisely, the Lebesgue spine is often associated with a particular subset of the Euclidean space that is built by taking a measurable set and considering a family of "spines" or "slices" that cover it.

Multipole expansion is a mathematical technique used in physics and engineering to simplify the description of a distribution of charge or mass, particularly in the context of fields generated by such distributions, like electric and gravitational fields. It is especially useful when the observation point is far from the source distribution, allowing for an approximation that captures the essential features of the field generated by the source.

The Newtonian potential, also known as the gravitational potential, describes the gravitational field generated by a mass distribution in classical physics. It is derived from Newton's law of universal gravitation and provides a way to calculate the gravitational potential energy per unit mass at a given point in space due to a mass or a distribution of mass.

The Perron method typically refers to techniques associated with the Perron-Frobenius theorem in the context of linear algebra and the study of non-negative matrices and certain types of dynamical systems. The theorem has important implications in various fields, such as economics, graph theory, and the study of Markov chains.

A **pluripolar set** is a concept in several complex variables and complex geometry. It arises in the context of pluripotential theory, which studies functions of several complex variables and their properties. In simple terms, a set \( E \) in \( \mathbb{C}^n \) (the n-dimensional complex space) is called pluripolar if it is contained in the set where a plurisubharmonic function is non-positive.

The Poisson kernel is a fundamental concept in harmonic analysis and potential theory, particularly in the study of solutions to the Laplace equation. It is used, among other things, to express the solution to the Dirichlet problem for the unit disk.

Polarization constants refer to specific values that characterize the degree and nature of polarization in a medium or system. In different contexts, the term can represent different concepts: 1. **In Electromagnetics**: Polarization constants can be associated with the polarization of electromagnetic waves. They may denote values that describe how the electric field vector of a wave is oriented in relation to the direction of propagation and how that orientation influences interactions with materials (like reflection, refraction, and absorption).

A potential energy surface (PES) is a conceptual and mathematical representation of the potential energy of a system, typically in the context of molecular and quantum mechanics. It describes how the potential energy of a system varies with the configuration of its particles, such as the positions of atoms in a molecule.

Quadrature domains are a mathematical concept related to the representation and computation of complex functions, particularly in the context of numerical analysis and function approximation. The term is often associated with the study of solutions to partial differential equations (PDEs) and can also be linked to various topics in analysis, such as potential theory and conformal mappings.

The Riesz potential is a generalization of the concept of the classical potential in mathematical analysis, particularly in potential theory and the study of fractional integrals. It is named after the mathematician Fritz Riesz.

The Riesz transform is a mathematical operator that is primarily used in the field of harmonic analysis and partial differential equations. Named after the mathematician Frigyes Riesz, it is associated with the study of functions defined on Euclidean spaces. In a more formal mathematical context, the Riesz transform can be defined in terms of the Laplace operator.

Spherical multipole moments are mathematical constructs used to describe the distribution of charge, mass, or other physical quantities in a system, particularly in the field of electromagnetism and gravitational fields. They extend the concept of electric or gravitational moments beyond the traditional dipole, quadrupole, and higher-order moments to capture more complex arrangements.

A **subharmonic function** is a real-valued function that satisfies specific mathematical properties, particularly within the context of harmonic analysis and the theory of partial differential equations.