Elliptic operator

An elliptic operator is defined via its principal symbol, which is a homogeneous polynomial obtained from the highest-order terms of the operator. For a linear differential operator of order m on a manifold, ellipticity requires that the symbol is nonzero for every nonzero cotangent vector. This condition ensures that the operator behaves like a generalized Laplacian at the level of leading symbols.

The definition extends to systems and pseudodifferential operators, where the principal symbol is a matrix or a symbol in the Hörmander class. Ellipticity is invariant under smooth coordinate changes, making it a well-defined geometric property.

Elliptic operators satisfy the fundamental property of elliptic regularity: if Lu is smooth, then u is smooth (up to the boundary in suitable settings). This is a consequence of the invertibility of the principal symbol, which allows the construction of a parametrix—an approximate inverse that is a pseudodifferential operator of order −m.

On compact manifolds without boundary, an elliptic operator is Fredholm: its kernel and cokernel are finite-dimensional, and its index (dimension of kernel minus dimension of cokernel) is a topological invariant. This index is the subject of the Atiyah–Singer index theorem, which expresses it in terms of characteristic classes.

Classical examples include the Laplacian, the Cauchy–Riemann operator, and the Dirac operator. The Laplacian on a Riemannian manifold is elliptic and its kernel consists of harmonic functions, which are constant on connected compact manifolds. The Dirac operator, arising in spin geometry, is elliptic and its index relates to the Â-genus.

Elliptic operators are central in geometric analysis, including Hodge theory, which uses the Laplacian on forms to decompose cohomology. They also appear in mathematical physics, e.g., in the study of the Seiberg–Witten equations, where ellipticity ensures the moduli space is smooth.

Ellipticity is not limited to scalar operators; systems like the Lamé operator in elasticity are elliptic, but the condition is more subtle for systems with variable coefficients. The concept also extends to subelliptic operators, where the symbol is only invertible along a subbundle, as in Hörmander's sum of squares.

Historically, the term 'elliptic' was borrowed from the classification of conic sections, reflecting the analogy with the Laplace equation. A lesser-known fact is that the index of an elliptic operator is stable under continuous deformations, which is the key to its topological nature. Also, on non-compact manifolds, ellipticity alone does not guarantee Fredholmness; additional conditions at infinity are required.

Principal symbol

The highest-order part of a differential operator, viewed as a polynomial on the cotangent bundle.

Fredholm operator

A bounded operator with finite-dimensional kernel and cokernel and closed range.

Parametrix

An approximate inverse of an elliptic operator, used to prove regularity.