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

For other uses, see Lacuna.

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.