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D. zn. G¨ ottingen (Math. Phys. ) (1918), 37-44. 44 [24] P. J. ”, Graduate Texts in Mathematics 107 Springer-Verlag, New York, 1993. [25] L. V. Ovsiannikov, “Group Analysis of Differential Equations”, Academic Press, New York, 1982. [26] K. R. Payne, Some remarks on the geometric structure underlying differential operators of mixed and degenerate types, in preparation. [27] Pohoˇzaev, S. I. On the eigenfunctions of the equation ∆u + λf (u) = 0. Soviet Math. Dokl. 6 (1965), 1408-1411. [28] Protter, M.

Mat. 9 (1945), 121-143. [8] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 338-418. [9] P. ”, Chelsea Publishing Co. New York, 1986. [10] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), 71-98. [11] S. Gellerstedt, Sur un probl`eme aux limites pour une ´equation lin´eaire aux d´eriv´ees partielles du second ordre de type mixtes, Dissertation, Uppsala University, Uppsala 1935.

Mixed signature metrics have received some attention in the context of quantum cosmology, beginning with the Hawking-Hartle no boundary hypothesis which if true implies that space-time evolved from a Riemannian past into a Lorentzian present, as well as models for tunnelling effects in quantum gravity (cf. Chapters 3 and 5 of [15]). 5 are strongly related to the conformal transformations with respect to the underlying geometric structure given above. 4). 21). 12) of the appendix. 5 will show that this part of the system is finite dimensional and spanned by the translations, dilations, and inversions above.

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Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types by Lupo D., Payne K.R.

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