Operator theory Ordinary and partial differential operators Concept
Elliptic operator
Summary
An elliptic operator is a linear differential operator whose principal symbol does not vanish at any nonzero covector. The Laplacian is the model example. Elliptic operators have strong regularity properties: solutions of are as smooth as the data allow. On compact manifolds they are Fredholm, which is the setting of index theory.
Where this leads
17 pages build on this one.
- Elliptic regularity
- Removable singularities for harmonic functions
- Minimum principle for harmonic functions
- Fredholm property of elliptic operators
- Lopatinskii–Shapiro condition
- Maximum principle
- Comparison principle
- Hopf lemma
- Schauder estimates
- Hypoelliptic operator
- Elliptic boundary value problem
- Gårding's inequality
- Maximum principle for harmonic functions
- Elliptic operator on a compact manifold
- Bôcher's theorem
- Morrey's inequality
- Dirac operator
Further reading
About this page
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| URI | m2://XY8GEQSWPR |
| Kind | concept |
| Writing | W0 stub |
| MSC | 47F10, 35J99 |
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Used by
- Elliptic regularity
- Lopatinskii–Shapiro condition
- Maximum principle
- Hopf lemma
- Schauder estimates
- Hypoelliptic operator
- Elliptic boundary value problem
- Gårding's inequality
- Elliptic operator on a compact manifold
- Dirac operator