Analysis

Learn Potential Theory

Harmonic functions past the classical tier: subharmonicity and Perron's method, capacity and energy, which boundary points a solution can reach, the fine topology, and the discrete and Brownian pictures of the same theory.

Free to start · adaptive placement finds your level · reviews timed to your own forgetting.

What you'll learn

30 lessons in Potential Theory

Subharmonic functionsHarnack's inequalityPerron's methodBarriers and regular boundary pointsThe Kelvin transformPotentials of measuresThe Riesz decompositionThe energy integral and the equilibrium measureCapacityTransfinite diameter and Chebyshev's constantPolar sets and Frostman's lemmaThinness and the fine topologyWiener's criterion and the Lebesgue spineHarmonic measureGreen's functions and Green potentialsBalayage and the obstacle problemPotential theory on networksBrownian motion, Kakutani, and the Martin boundaryNontangential limits and Fatou's theoremThe boundary Harnack principleThe Neumann problemSteklov eigenvalues and the Dirichlet-to-Neumann mapCalderón's inverse problemComputing capacitiesThe p-LaplacianThe fractional LaplacianParabolic potential theoryQuadrature domainsHele-Shaw flow and Laplacian growthDiffusion-limited aggregation
How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Potential Theory exactly where it's useful.

Related Analysis subjects