1 | What is potential theory? Basic properties of harmonic functions.
Using Brownian Motion to solve the Dirichlet problem. Recurrence and transience.
Convergence of bounded martingales. Further applications to complex analysis.
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2 |
The Poisson kernel. No positive harmonic functions in R^n. Harmonic functions are real analytic. Harnack's
principle.
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3 | Basic properties of subharmonic functions. Phragmen-Lindelof principle. Liouville's
theorem for subharmonic functions.
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4 | Distributions. Weyl's lemma. Riesz decomposition. Potentials. Energy. Equilibrium measures. Polar Sets.
Frostman's theorem. Brelot-Cartan theorem. Minus infinity sets. Removable singularities.
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5 | Capacity and transfinite diameter. Hilbert lemniscate theorem.
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6 | Schauder's fixed point theorem. Existence and uniqueness of solutions of the Gauss curvature equation with prescribed boundary data.
Liouville's theorem on the correspondence between conformal metrics of curvature -4 and holomorphic self-maps of the unit disk. Heins theorem on prescribing finite Blaschke products by their critical points.
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7 | Basic properties of Sobolev spaces. Regularity of solutions of the linear Dirichlet problem with singularities.
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8 | Maximal functions. Sobolev capacities. Removable sets for Sobolev functions.
Obstacle problems. Carleson's theorem on removable sets of Holder continuous harmonic functions.
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