Hypoellipticity

In the theory of partial differential equations, a partial differential operator P {\displaystyle P} defined on an open subset

U ⊂ R n {\displaystyle U\subset {\mathbb {R} }^{n}}

is called hypoelliptic if for every distribution u {\displaystyle u} defined on an open subset V ⊂ U {\displaystyle V\subset 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). In addition, the operator for the heat equation ( P ( u ) = u t − k Δ u {\displaystyle P(u)=u_{t}-k\,\Delta u\,} )

P = ∂ t − k Δ x {\displaystyle P=\partial _{t}-k\,\Delta _{x}\,}

(where k > 0 {\displaystyle k>0} ) is hypoelliptic but not elliptic. However, the operator for the wave equation ( P ( u ) = u t t − c 2 Δ u {\displaystyle P(u)=u_{tt}-c^{2}\,\Delta u\,} )

P = ∂ t 2 − c 2 Δ x {\displaystyle P=\partial _{t}^{2}-c^{2}\,\Delta _{x}\,}

(where c ≠ 0 {\displaystyle c\neq 0} ) is not hypoelliptic.

References

• Shimakura, Norio (1992). Partial differential operators of elliptic type: translated by Norio Shimakura. American Mathematical Society, Providence, R.I. ISBN 0-8218-4556-X.
• Egorov, Yu. V.; Schulze, Bert-Wolfgang (1997). Pseudo-differential operators, singularities, applications. Birkhäuser. ISBN 3-7643-5484-4.
• Vladimirov, V. S. (2002). Methods of the theory of generalized functions. Taylor & Francis. ISBN 0-415-27356-0.
• Folland, G. B. (2009). Fourier Analysis and its applications. AMS. ISBN 978-0-8218-4790-9.

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