Elliptic operator

A differential operator whose principal symbol is invertible at every nonzero cotangent vector. This condition underlies strong regularity results for solutions of its equations.

Principal symbol: The highest-order part of a differential operator, represented as a function on cotangent vectors. Its invertibility away from the zero covector is the defining test for ellipticity.
Differential operator: A linear or nonlinear operator built from derivatives of a function and coefficient functions. Elliptic operators are a structurally distinguished class of differential operators.
Hyperbolic operator: A differential operator whose principal symbol has a characteristic geometry associated with propagation of waves. Unlike elliptic equations, hyperbolic equations admit characteristic directions and wave propagation.
Poisson equation: The equation Δu = f, relating a function's Laplacian to a prescribed source. It is a central inhomogeneous elliptic equation in physics and mathematics.
Elliptic system: A coupled system of differential equations satisfying an ellipticity condition on its combined principal symbol. Systems require symbol-level conditions that are subtler than scalar ellipticity.
Elliptic regularity: The principle that elliptic equations force solutions to be smoother than their initial data may suggest. Ellipticity converts weak control of a solution into stronger smoothness.
Cotangent space: The vector space of linear functionals on a tangent space, whose elements represent covectors. Ellipticity tests the principal symbol on nonzero covectors.
Parabolic operator: A differential operator associated with diffusion-type evolution and a distinguished time direction. Parabolic operators combine spatial ellipticity with evolution in time.
Laplace equation: The equation Δu = 0, whose solutions are called harmonic functions. Its solutions exemplify the smoothness and mean-value behavior of elliptic equations.
Strong ellipticity: A strengthened ellipticity condition that supplies coercive estimates in many systems and boundary problems. It often provides the estimates needed for robust existence and spectral results.

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