31Axx Two-dimensional potential theory
This subtopic studies two-dimensional potential theory, focusing on harmonic functions, Green functions, and boundary behavior in the plane.
Specific topics
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
Overview
31A05 studies harmonic, subharmonic, superharmonic functions in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for harmonic, subharmonic, superharmonic functions in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks
31A10 Integral representations, integral operators, integral equations methods in two dimensions
Overview
31A10 studies integral representations, integral operators, integral equations methods in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for integral representations, integral operators, integral equations methods in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks
31A15 Potentials and capacity, harmonic measure, extremal length in two dimensions
Overview
31A15 studies potentials and capacity, harmonic measure, extremal length in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for potentials and capacity, harmonic measure, extremal length in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks
31A20 Boundary behavior (theorems of Fatou type, etc.) in two dimensions
Overview
31A20 studies boundary behavior (theorems of fatou type, etc.) in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for boundary behavior (theorems of fatou type, etc.) in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks
31A25 Boundary value and inverse problems in two dimensions
Overview
31A25 studies boundary value and inverse problems in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for boundary value and inverse problems in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks
31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation in two dimensions
Overview
31A30 studies biharmonic, polyharmonic functions and equations, poisson's equation in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for biharmonic, polyharmonic functions and equations, poisson's equation in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks
31A35 Connections with differential equations in two dimensions
Overview
31A35 studies connections with differential equations in two dimensions in two-dimensional potential theory. It studies harmonic, subharmonic, and Green-theoretic phenomena in the plane, emphasizing capacity, boundary behavior, and logarithmic potentials.
Related Wikipedia Page
Two-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for connections with differential equations in two dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to analyze harmonic measure, Dirichlet problems, and conformal invariants in the plane.
Applications
- Planar harmonic analysis
- Dirichlet and boundary value problems
- Capacity and Green function methods
References
Recommended Textbooks