Potential Theory and Differential Equations Analysis

Potential theory lies at the intersection of analysis and geometry, concerned with harmonic, subharmonic and superharmonic functions, and the associated notions of capacity, equilibrium measures and energy integrals. Its origins in electrostatics and gravitational fields have expanded to embrace a broad spectrum of boundary‐value problems for Laplace’s equation and its generalisations. Differential equations analysis provides the language and tools to describe how these potentials evolve, whether as steady‐state solutions of elliptic problems, time‐dependent solutions of parabolic equations or propagating waves in hyperbolic systems. Central techniques include Green’s functions, variational principles and maximum‐principle arguments, which together facilitate the resolution of classical and singular boundary conditions in Euclidean and curved spaces. Recent advances have explored nonlocal operators, degenerate and singular coefficients, geometric settings modelled by the Laplace–Beltrami operator and fractional processes. Applications range from fluid flow and electromagnetic scattering to financial mathematics and image processing. The vibrant interplay of analytic, geometric and computational methods continues to yield new insights into regularity theory, spectral properties and inverse problems, underscoring the enduring significance of potential theory in contemporary analysis.

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Potential Theory and Differential Equations Analysis publication trend

The graph below shows the total number of articles in potential theory and differential equations analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Green’s function: A kernel that represents the influence at one point due to a unit source at another, used to solve linear boundary-value problems.

Quadrature domain: A region in which certain weighted volume integrals of harmonic or Helmholtz solutions can be evaluated exactly by a finite sum of pointwise values.

Caloric measure: A probability measure on the boundary of a domain that describes the distribution of heat emanating from boundary points for the heat equation.

Fundamental solution: A function that represents the response of a differential operator to a point source, serving as a building block for general solutions.

Laplace–Beltrami operator: The generalisation of the Laplacian to curved manifolds, incorporating the metric structure of the underlying space.

References

1. Multi-phase k-quadrature domains and applications to acoustic waves and magnetic fields. Partial Differential Equations and Applications (2024).
2. The set of mildly regular boundary points has full caloric measure. Transactions of the London Mathematical Society (2023).
3. Fundamental Solutions for the Laplace–Beltrami Operator Defined by the Conformal Hyperbolic Metric and Jacobi Polynomials. Complex Analysis and Operator Theory (2023).