Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

35Hxx Close-to-elliptic equations

This subtopic studies PDEs close to elliptic equations, where near-ellipticity guides regularity, solvability, and perturbation arguments.

Specific topics

35H10 Hypoelliptic equations

Overview

35H10 treats hypoelliptic equations within close-to-elliptic equations. The focus is on model classes, principal estimates, and solution mechanisms that clarify existence, uniqueness, regularity, and asymptotic behavior in representative PDE settings.

Related Wikipedia Page

Wikipedia search: Hypoelliptic equations

Useful Links

Key Ideas

  • Canonical formulations and boundary or initial-value settings for hypoelliptic equations
  • Regularity, well-posedness, and qualitative behavior of solutions
  • Functional-analytic and microlocal tools used to derive estimates and structure results

Typical Uses

Used to classify PDE models, select analytic techniques, and build proof strategies for existence, regularity, and stability results in both pure and applied contexts.

Applications

  • Mathematical physics and continuum models involving diffusion, waves, or transport
  • Geometric and variational PDE problems where existence and stability are central
  • Numerical analysis workflows that require rigorous PDE regularity and estimate frameworks

References

Recommended Textbooks

35H20 Subelliptic equations

Overview

35H20 treats subelliptic equations within close-to-elliptic equations. The focus is on model classes, principal estimates, and solution mechanisms that clarify existence, uniqueness, regularity, and asymptotic behavior in representative PDE settings.

Related Wikipedia Page

Wikipedia search: Subelliptic equations

Useful Links

Key Ideas

  • Canonical formulations and boundary or initial-value settings for subelliptic equations
  • Regularity, well-posedness, and qualitative behavior of solutions
  • Functional-analytic and microlocal tools used to derive estimates and structure results

Typical Uses

Used to classify PDE models, select analytic techniques, and build proof strategies for existence, regularity, and stability results in both pure and applied contexts.

Applications

  • Mathematical physics and continuum models involving diffusion, waves, or transport
  • Geometric and variational PDE problems where existence and stability are central
  • Numerical analysis workflows that require rigorous PDE regularity and estimate frameworks

References

Recommended Textbooks

35H30 Quasi-elliptic equations

Overview

35H30 treats quasi-elliptic equations within close-to-elliptic equations. The focus is on model classes, principal estimates, and solution mechanisms that clarify existence, uniqueness, regularity, and asymptotic behavior in representative PDE settings.

Related Wikipedia Page

Wikipedia search: Quasi-elliptic equations

Useful Links

Key Ideas

  • Canonical formulations and boundary or initial-value settings for quasi-elliptic equations
  • Regularity, well-posedness, and qualitative behavior of solutions
  • Functional-analytic and microlocal tools used to derive estimates and structure results

Typical Uses

Used to classify PDE models, select analytic techniques, and build proof strategies for existence, regularity, and stability results in both pure and applied contexts.

Applications

  • Mathematical physics and continuum models involving diffusion, waves, or transport
  • Geometric and variational PDE problems where existence and stability are central
  • Numerical analysis workflows that require rigorous PDE regularity and estimate frameworks

References

Recommended Textbooks