By Braides A., Malchiodi A.

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H. Ibragimov. ), Boca Raton, CRC Press, 1995 9. H. Ibragimov, Elementary Lie groups analysis and ODE's, New York, Wiley 1999 10. L. Dresner, Applications of Lie's theory of ordinary and partial differential equations, Bristol, Institut of Physics, 1999 11. E. Hydon, Symmetry methods for differential equations, Cambridge, Cambridge University Press, 2000 12. J. Olver and Ph. Rosenau, Phys. Lett. A, 114, 107 (1986) and SIAM J. Appl. Math. 47, 263 (1987) 13. D. Levi and P. Winternitz, J. Phys. A 22, 2915 (1989) and J.

163-184 (1993). 4. Benenti, J. Math. Phys. 38 (12), 6578-6602 (1997). 5. Rastelli, J. Math. Phys. 41 (7), 4654-4678 (2000). 6. Rastelli, J. Math. Phys. 42 (5), 2065-2091 (2001). 7. Rastelli, "Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrodinger equation. 1. The completeness and the Robertson conditions" J. Math. Phys. 43 (10), (2002)(forthcoming). 8. Eisenhart, Ann. Math. 35, 284-305 (1934). 9. Levi-Civita, Math.

PARTIAL SYMMETRIES A N D S Y M M E T R I C SETS OF SOLUTIONS TO PDE'S GIAMPAOLO CICOGNA Dip. it sez. e. of sets which are mapped into themselves by the symmetry. We introduce the definition of partial symmetry, which in some sense generalizes the concept of conditional symmetry, and show that the above notions may be preserved, although the symmetry is not exact. Some simple examples are also considered. 1. I n t r o d u c t i o n The usefulness of the notion of symmetry of a differential equation certainly does not need to be emphasized, both for its applications in the abstract analysis, and for the very concrete aim of finding explicit solutions (see 1 - u and references therein); but probably the most obvious and direct consequence is the fact that any symmetry of a given equation transforms solutions into (generally, different) solutions of the same equation.

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Curvature theory of boundary phases the two-dimensional case by Braides A., Malchiodi A.


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