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Exotic n-D’Alembert PDEs and Stability

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Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 68))

Abstract

In the framework of the PDE’s algebraic topology, previously introduced by A. Prástaro, exotic n-d’Alembert PDEs are considered. These are n-d’Alembert PDEs, (d′A) n , admitting Cauchy manifolds N⊂(d′A) n identifiable with exotic spheres, or such that ∂N can be exotic spheres. For such equations, local and global existence theorems and stability theorems are obtained. (See also Prástaro in arXiv:1011.0081, 2010.)

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Notes

  1. 1.

    For general information on bordism groups and related problems in differential topology and PDE’s geometry, see, e.g., [5, 9–13, 15, 22–29, 31, 47, 48, 50–53]. For crystallographic groups, see the references quoted in [37]. For differential structures and algebraic topology of exotic spheres, see [6–8, 16–21, 35, 39–41, 46].

  2. 2.

    Here by the integral bordism group we mean the weak integral bordism group \(\varOmega^{E_{k}}_{n-1,w}\).

  3. 3.

    Recall that A≡Map(Ω,ℝ), Ω a group, has a natural structure of a Hopf algebra if Ω is a finite group. If Ω is not finite, then A has a structure of a Hopf algebra in an extended sense. (See [25].)

  4. 4.

    An extended 0-crystal PDE \(E_{k}\subset J^{k}_{n}(W)\) is not necessarily a 0-crystal PDE. In fact, in order for E k to be an extended 0-crystal PDE it is enough that \(\varOmega _{n-1,w}^{E_{k}}=0\). This does not necessarily imply that \(\varOmega_{n-1}^{E_{k}}=0\).

  5. 5.

    In this paper, we will use the same notation adopted in [40]: \(\thickapprox\) homeomorphism; ≅ diffeomorphism; ≊ homotopy equivalence; ≃ homotopy.

  6. 6.

    If n=2 we simply say d’Alembert equation and we will put (d′A)≡(d′A)2.

  7. 7.

    For example, for n=2 one has F=u xy u−u x u y , and for n=3 one has F=u xyz u 2−u xy u z u−u xz u y u+u x u y u z .

  8. 8.

    Θ n denotes the additive group of diffeomorphism classes of oriented smooth homotopy spheres of dimension n.

  9. 9.

    Let us emphasize that to Ω[V] belong also (not necessarily regular) solutions V′⊂E k such that \(N_{0}'\sqcup N_{1}'=N_{0}\sqcup N_{1}\), where \(\partial V'=N_{0}'\bigcup P'\bigcup N_{1}'\).

  10. 10.

    In the following, if there are no reasons for confusion, we shall also call a stable solution a smooth regular solution of a PDE E k ⊂JD k(W) that is average asymptotic stable.

  11. 11.

    τ 0 has just the physical dimension of a time.

  12. 12.

    (d′A) n considered in this theorem is a submanifold of JD n(E), hence it coincides with Z n ∩C n .

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Acknowledgements

I would like to thank Editors for their kind invitation to contribute my paper to this book, dedicated to Themistocles M. Rassias on occasion of his 60th birthday.

Work partially supported by MIUR Italian grants “PDE’s Geometry and Applications”.

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Authors and Affiliations

  1. MEMOMAT, University of Roma “La Sapienza”, Via A. Scarpa, 16, 00161, Roma, Italy

    Agostino Prástaro

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  1. Agostino Prástaro
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Correspondence to Agostino Prástaro .

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Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. Center for Applied Optimization, Dept. Industrial & Systems, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Pando G. Georgiev

  3. , Department of Mathematics & Statistics, University of Victoria, Finnerty Road 3800, Victoria, V8W 3R4, British Columbia, Canada

    Hari M. Srivastava

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Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.

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Prástaro, A. (2012). Exotic n-D’Alembert PDEs and Stability. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_36

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