This volume studies harmonic, subharmonic, and superharmonic functions, along with potentials and their applications to analysis, geometry, and...
This volume studies harmonic, subharmonic, and superharmonic functions, along with potentials and their applications to analysis, geometry, and physics. It connects closely with partial differential equations and complex analysis.
Part I. Foundations
Chapter 1. Harmonic Functions
1.1 Definition via Laplace equation 1.2 Mean value property 1.3 Maximum and minimum principles 1.4 Uniqueness results 1.5 Examples
Chapter 2. Subharmonic and Superharmonic Functions
2.1 Definitions 2.2 Basic properties 2.3 Comparison principles 2.4 Upper and lower envelopes 2.5 Examples
Chapter 3. Laplace Equation
3.1 Formulation 3.2 Solutions in ℝⁿ 3.3 Boundary value problems 3.4 Uniqueness and existence 3.5 Examples
Part II. Classical Potential Theory
Chapter 4. Newtonian Potentials
4.1 Fundamental solutions 4.2 Potential functions 4.3 Properties 4.4 Applications 4.5 Examples
Chapter 5. Green’s Functions
5.1 Definition 5.2 Construction 5.3 Boundary conditions 5.4 Applications 5.5 Examples
Chapter 6. Dirichlet Problem
6.1 Formulation 6.2 Existence and uniqueness 6.3 Perron method 6.4 Applications 6.5 Examples
Part III. Energy and Capacity
Chapter 7. Energy Integrals
7.1 Definition 7.2 Properties 7.3 Minimization principles 7.4 Applications 7.5 Examples
Chapter 8. Capacity
8.1 Definition 8.2 Properties 8.3 Relation to measure 8.4 Applications 8.5 Examples
Chapter 9. Equilibrium Measures
9.1 Definition 9.2 Existence and uniqueness 9.3 Properties 9.4 Applications 9.5 Examples
Part IV. Fine Properties
Chapter 10. Fine Topology
10.1 Definition 10.2 Fine continuity 10.3 Thin sets 10.4 Applications 10.5 Examples
Chapter 11. Boundary Behavior
11.1 Regular boundary points 11.2 Fatou-type theorems (overview) 11.3 Limits at boundary 11.4 Applications 11.5 Examples
Chapter 12. Singularities
12.1 Isolated singularities 12.2 Removable singularities 12.3 Classification 12.4 Applications 12.5 Examples
Part V. Probabilistic Connections
Chapter 13. Brownian Motion
13.1 Definition 13.2 Connection to harmonic functions 13.3 Hitting probabilities 13.4 Applications 13.5 Examples
Chapter 14. Stochastic Processes
14.1 Markov processes 14.2 Potential kernels 14.3 Probabilistic representation 14.4 Applications 14.5 Examples
Chapter 15. Dirichlet Forms
15.1 Definitions 15.2 Energy interpretation 15.3 Connections to PDEs 15.4 Applications 15.5 Examples
Part VI. Potential Theory in Complex Analysis
Chapter 16. Logarithmic Potentials
16.1 Definitions 16.2 Properties 16.3 Equilibrium problems 16.4 Applications 16.5 Examples
Chapter 17. Plurisubharmonic Functions (Overview)
17.1 Definitions 17.2 Properties 17.3 Complex potential theory 17.4 Applications 17.5 Examples
Chapter 18. Capacity in Complex Analysis
18.1 Definitions 18.2 Relations to sets 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. PDE Applications
19.1 Elliptic equations 19.2 Boundary value problems 19.3 Regularity 19.4 Applications 19.5 Examples
Chapter 20. Geometry Applications
20.1 Minimal surfaces (overview) 20.2 Geometric flows 20.3 Potential methods 20.4 Applications 20.5 Examples
Chapter 21. Physics Applications
21.1 Electrostatics 21.2 Gravitational fields 21.3 Fluid flow 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Nonlinear potential theory 22.2 Metric space potential theory 22.3 Connections to analysis 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Boundary regularity 23.2 Capacity estimates 23.3 Nonlinear equations 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of potential theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Fundamental solutions reference B. Key theorems summary C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume develops potential theory as the study of harmonic behavior and energy minimization. It connects analysis, probability, and physics through the Laplace equation and related structures.