31Bxx Higher-dimensional potential theory
This subtopic studies higher-dimensional potential theory, extending classical harmonic and potential methods to spaces of larger dimension.
Specific topics
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
Overview
31B05 studies harmonic, subharmonic, superharmonic functions in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for harmonic, subharmonic, superharmonic functions in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
Overview
31B10 studies integral representations, integral operators, integral equations methods in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for integral representations, integral operators, integral equations methods in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
Overview
31B15 studies potentials and capacities, extremal length and related notions in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for potentials and capacities, extremal length and related notions in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks
31B20 Boundary value and inverse problems in higher dimensions
Overview
31B20 studies boundary value and inverse problems in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for boundary value and inverse problems in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks
31B25 Boundary behavior of harmonic functions in higher dimensions
Overview
31B25 studies boundary behavior of harmonic functions in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for boundary behavior of harmonic functions in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
Overview
31B30 studies biharmonic and polyharmonic equations and functions in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for biharmonic and polyharmonic equations and functions in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks
31B35 Connections with differential equations in higher dimensions
Overview
31B35 studies connections with differential equations in higher dimensions in higher-dimensional potential theory. It studies harmonic and subharmonic phenomena in higher dimensions, including Green functions, capacity, and boundary regularity.
Related Wikipedia Page
Higher-Dimensional Potential Theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for connections with differential equations in higher dimensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in elliptic PDE, probabilistic potential theory, and geometric analysis.
Applications
- Green functions and capacity
- Higher-dimensional harmonic analysis
- Probabilistic potential theory
References
Recommended Textbooks