Equations with constant coefficients invariant under a group of linear transformations
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- by André Cerezo PDF
- Trans. Amer. Math. Soc. 204 (1975), 267-298 Request permission
Abstract:
If $P$ is a linear differential operator on ${{\mathbf {R}}^n}$ with constant coefficients, which is invariant under a group $G$ of linear transformations, it is not true in general that the equation $Pu = f$ always has a $G$-invariant solution $u$ for a $G$-invariant $f$. We elucidate here the particular case of a “big” group $G$, and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.References
- M. F. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145–150. MR 256156, DOI 10.1002/cpa.3160230202
- Reese Harvey, Hyperfunctions and linear partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 1042–1046. MR 200604, DOI 10.1073/pnas.55.5.1042
- Hikosaburo Komatsu, An introduction to the theory of hyperfunctions, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973, pp. 3–40. MR 0394190
- Pierre-Denis Methée, Sur les distributions invariantes dans le groupe des rotations de Lorentz, Comment. Math. Helv. 28 (1954), 225–269 (French). MR 64268, DOI 10.1007/BF02566932
- Tetsuji Miwa, On the existence of hyperfunction solutions of linear differential equations of the first order with degenerate real principal symbols, Proc. Japan Acad. 49 (1973), 88–93. MR 348236
- Mustapha Raïs, Solutions élémentaires des opérateurs différentiels bi-inbariants sur un groupe de Lie nilpotent, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A495–A498 (French). MR 289720
- Marcel Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), 1–223 (French). MR 30102, DOI 10.1007/BF02395016
- Mikio Sato, Theory of hyperfunctions. I, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1959), 139–193. MR 114124
- Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973, pp. 265–529. MR 0420735 P. Shapira, Théorie des hyperfonctions, Lecture Notes in Math., vol. 126, Springer-Verlag, Berlin and New York, 1970.
- Georges de Rham, Sur la division de formes et de courants par une forme linéaire, Comment. Math. Helv. 28 (1954), 346–352 (French). MR 65241, DOI 10.1007/BF02566941
- Pierre-Denis Methée, Transformées de Fourier de distributions invariantes. II, C. R. Acad. Sci. Paris 241 (1955), 684–686 (French). MR 76093
- Georges de Rham, Solution élémentaire d’opérateurs différentiels du second ordre, Ann. Inst. Fourier (Grenoble) 8 (1958), 337–366 (French). MR 117437, DOI 10.5802/aif.83
- L. Gȧrding and J.-L. Lions, Functional analysis, Nuovo Cimento (10) 14 (1959), no. supplemento, 9–66. MR 117543, DOI 10.1007/BF03026447
- A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201–218. MR 126154, DOI 10.7146/math.scand.a-10610
- Raïs Mustapha, Les solutions invariantes de l’équation des ondes, C. R. Acad. Sci. Paris 259 (1964), 2169–2170 (French). MR 173080
- Viggo Edén, Distributions invariant under the group of complex orthogonal transformations, Math. Scand. 14 (1964), 75–89. MR 173944, DOI 10.7146/math.scand.a-10707
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 204 (1975), 267-298
- MSC: Primary 35E99; Secondary 46F15, 58G15
- DOI: https://doi.org/10.1090/S0002-9947-1975-0430501-1
- MathSciNet review: 0430501