Atiyah, Michael F.; Bott, Raoul; Gårding, Lars Lacunas for hyperbolic differential operators with constant coefficients.I. (English) Zbl 0191.11203 Acta Math. 124, 109-189 (1970). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 7 ReviewsCited in 91 Documents MathOverflow Questions: Is there a formulation of Huygens’ principle using the language of algebraic geometry? PDEs and algebraic varieties Keywords:partial differential equations × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Atiyah, M. F. &Hodge, W. D. V., Integrals of the second kind of an algebraic variety.Ann. of Math., 62 (1955), 56–91. · Zbl 0068.34401 · doi:10.2307/2007100 [2] Bazer, J. &Yen, D. H. Y., The Riemann matrix of (2+1)-dimensional symmetric-hyperbolic systems.Comm. Pure Appl. Math., 20 (1967), 329–363. · Zbl 0163.33601 [3] Bazer, J. & Yen, D. H. Y.,Lacunas of the Riemann matrix of symmetric-hyperbolic systems in two space variables. Preprint. Courant Institute, New York University, 1969. · Zbl 0167.10003 [4] Borovikov, V. A., The elementary solution of partial differential equations with constant coefficients.Trudy Moskov. Mat. Obšč., 66, 8 (1959), 159–257 [5] –, Some sufficient conditions for the absence of lacunas.Mat. Sb., 55 (97) (1961), 237–254. [6] Burridge, R., Lacunas in two-dimensional wave propagation.Proc. Cambridge Phil. Soc., 63 (1967), 819–825. · Zbl 0189.26502 · doi:10.1017/S0305004100041803 [7] Gårding, L., The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.Ann. of Math., 48 (1947), 785–826. Errataibid. Gårding, L., The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.Ann. of Math., 52 (1950), 506–507. · Zbl 0029.21601 · doi:10.2307/1969381 [8] –, Linear hyperbolic partial differential equations with constant coefficients.Acta Math., 85 (1950), 1–62. · Zbl 0045.20202 · doi:10.1007/BF02395740 [9] –, An inequality for hyperbolic polynomials.J. Math. Mech., 8 (1959), 957–966. · Zbl 0090.01603 [10] –, Transformation de Fourier des distributions homogènes.Bull. Soc.math. France, 89 (1961) 381–428. · Zbl 0102.32501 [11] Gårding, L.,The theory of lacunas. Battelle Seattle 1968 Recontres. Springer (1969). · Zbl 0191.11301 [12] Gelfand, I. M. & Shilov, G. E.,Generalized functions I. Moscow 1958. [13] Gindikin, S. G., Cauchy’s problem for strongly homogeneous differential operators.Trudy Moskov. Mat. Obšč., 16 (1967), 181–208. · Zbl 0194.13102 [14] Gindikin, S. G. &Vajnberg, B. R., On a strong form of Huygens’ principle for a class of differential operators with constant coefficients.Trudy Moskov. Mat. Obšč., 16 (1967), 151–180. · Zbl 0194.12804 [15] Grothendieck, A., On the de Rham cohomology of algebraic varities.Publ. IHES, 29 (1966), 351–359. · Zbl 0145.17602 [16] Hadamard, J.,Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Paris 1932. · JFM 58.0519.16 [17] Herglotz, G., Über die Integration linearer partieller Differentialgleichungen I. (Anwendung Abelscher Integrale) II, III (Anwendung Fourierscher Integrale).Leipzig. Ber. Sächs. Akad. Wiss., Math Phys. Kl., 78 (1926), 93–126; 80 (1928), 6–114. [18] Hörmander, L.,Linear partial differential operators. Springer 1963. · Zbl 0108.09301 [19] Hörmander, L., On the singularities of solutions of partial differential equations.International Conference of Functional Analysis and Related Topics. Tokyo 1969. [20] Leray, J., Un prolongement de la transformation de Laplace... (Problème de Cauchy IV).Bull. Soc. math. France, 90 (1962), 39–156. · Zbl 0185.34302 [21] –,Hyperbolic differential equations. The Institute for Advanced Study, Princeton N. J. (1952). [22] Ludwig, D., Singularities of superpositions of distributions.Pacific J. Math. 15 (1965), 215–239. · Zbl 0143.20803 [23] Nuij, W., A note on hyperbolic polynomials.Math. Scand., 23 (1968), 69–72. · Zbl 0189.40803 [24] Petrovsky, I. G., On the diffusion of waves and the lacunas for hyperbolic equations.Mat. Sb., 17 (59) (1945), 289–370. · Zbl 0061.21309 [25] Riesz, M., L’intégrale de Riemann-Liouville et le problème de Cauchy.Acta Math., 81 (1949), 1–223. · Zbl 0033.27601 · doi:10.1007/BF02395016 [26] Schwartz, L.,Théorie des distributions I, II. Paris (1950–51). [27] Stellmacher, K. L., Eine Klasse huygenscher Differentialgeleichungen und ihre Integration.Math. Ann., 130 (1955), 219–233. · Zbl 0134.31101 · doi:10.1007/BF01343350 [28] Svensson, L., Necessary and sufficient conditions for the hyperbolicity of polynomials with hyperbolic principal part. To be published inArk. Mat., 8 (1970). · Zbl 0203.40903 [29] Weitzner, H., Green’s function for two-dimensional magnetohydrodynamic waves I, II.Phys. Fluids, 4 (1961), 1250–1258. · Zbl 0121.44601 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.