Elliptic operator

An elliptic operator is a type of linear differential operator whose principal symbol is invertible everywhere except at the zero covector. This condition — ellipticity — ensures that the operator behaves like a generalized Laplacian: solutions to elliptic equations are as smooth as the coefficients permit, and the operator has a finite-dimensional kernel and cokernel. The Laplace operator Δ = ∂²/∂x₁² + ... + ∂²/∂xₙ² is the prototypical example.

Ellipticity is the key hypothesis of the Atiyah-Singer index theorem, which computes the difference dim ker D − dim coker D in topological terms. Without ellipticity, this difference is typically infinite and uncomputable. The condition thus marks the boundary between "tame" and "wild" differential equations on manifolds.

The theory of elliptic operators extends beyond compact manifolds through the study of pseudodifferential operators, which generalize differential operators while preserving the essential spectral and regularity properties.