Hypoelliptic operator

In the theory of partial differential equations, a partial differential operator P {displaystyle P} defined on an open subset In the theory of partial differential equations, a partial differential operator P {displaystyle P} defined on an open subset is called hypoelliptic if for every distribution u {displaystyle u} defined on an open subset V ⊂ U {displaystyle Vsubset U} such that P u {displaystyle Pu} is C ∞ {displaystyle C^{infty }} (smooth), u {displaystyle u} must also be C ∞ {displaystyle C^{infty }} . If this assertion holds with C ∞ {displaystyle C^{infty }} replaced by real analytic, then P {displaystyle P} is said to be analytically hypoelliptic. Every elliptic operator with C ∞ {displaystyle C^{infty }} coefficients is hypoelliptic. In particular, the Laplacian is an example of a hypoelliptic operator (the Laplacian is also analytically hypoelliptic). The heat equation operator (where k > 0 {displaystyle k>0} ) is hypoelliptic but not elliptic. The wave equation operator (where c ≠ 0 {displaystyle c eq 0} ) is not hypoelliptic. This article incorporates material from Hypoelliptic on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.