Lectures on Potential Theory

I Introduction And Topological Preliminaries Introduction II General Capacities of Choquet and Capacitability Capacitability True capacity and capacitability Weak and Strong Capacity Weak capacity Properties of Inner and Outer capacities Strong Subadditivity Strong capacity Fundamental Theorem III Potentials with Kernels - Convergence Theorems Preliminaries on Measures, Kernels and Potentiels Radon Measures Radon Integrals of functions (for positive measures) Kernels and Potentials Negligible Sets and Regular Kernels Definitions and Fundamental Lemmas Associated kernel and Energy (Choquet) Examples of Regular kernels Convergence theorems (with exceptional G*-negligible sets)1 Case of a compact space Extension of the converge... G-capacity1 G-capacity and G-capacity measures G-capacity and G-negligible sets G-capacity and strong sub-additivity G-polar sets Second Group of Convergence Theorems Convergence Theorems An Application to The Balayage Principle1 IV Axiomatic Theory of Harmonic... Generalised Harmonic Functions The Fundamental axioms Examples First consequences The case where the constants are harmonic h-harmonic functions Superharmonic and Hyperharmonic Functions Minimum principle Local criterion Elementary case of ``balayage'' Nearly Hyper or Superharmonic Functions... Applications Sets of Harmonic and Superharmonic Functions... Saturated set of hyperharmonic functions Examples Harmonic minorants and majorants Lattice on harmonic functions Lattice on superharmonic functions Vector space of differences of... Potentials Existence of a Potentials > 0: Consequences General Dirichlet Problem Fundamental envelopes Properties of f Resolutivity Absolute Resolutivity Some Examples of Dirichlet Problem Second Example... Extension to a relatively compact open set Behaviour of the generalised solution... Third Example of Dirichlet Problem Reduced Functions and Polar Sets Polar sets Criterion Convergence Theorems Domination Principle First application of axiom D Convergence theorems Application to reduced functions Thins Sets - Some Applications1 Fine topology Further development Generalisation of the Riesz -Martin Integral Representation Integral representation of... Further Theory-Kernel Additional Chapter and Bibliography for Part IV...