Petrovsky lacuna

Petrovsky lacunas are similar to the spaces between shock waves of a supersonic object.

For other uses, see Lacuna.

In mathematics, a Petrovsky lacuna, named for the Russian mathematician I. G. Petrovsky, is a region where the fundamental solution of a linear hyperbolic partial differential equation vanishes. They were studied by Petrovsky (1945) who found topological conditions for their existence.

Petrovsky's work was generalized and updated by Atiyah, Bott, and Gårding (1970, 1973).

References

• Atiyah, Michael Francis (1966–1968), "Hyperbolic differential equations and algebraic geometry (after Petrowsky)", Séminaire Bourbaki, Vol. 10, Paris: Société Mathématique de France, pp. 87–99, MR 1610456, Zbl 0201.12501.
• Atiyah, Michael Francis; Bott, Raoul; Gårding, Lars (1970), "Lacunas for hyperbolic differential operators with constant coefficients. I", Acta Mathematica, 124: 109–189, doi:10.1007/BF02394570, MR 0470499, Zbl 0191.11203.
• Atiyah, Michael Francis; Bott, Raoul; Gårding, Lars (1973), "Lacunas for hyperbolic differential operators with constant coefficients. II", Acta Mathematica, 131: 145–206, doi:10.1007/BF02392039, MR 0470500, Zbl 0266.35045.
• Petrovsky, I.G. (1945), "On the diffusion of waves and the lacunas for hyperbolic equations", Recueil Mathématique (Matematicheskii Sbornik), 17 (59) (3): 289–368, MR 0016861, Zbl 0061.21309.