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Light Near a Caustic

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The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2043))

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Abstract

In the cold plasma model the sonic curve is a parabola. In the physical model presented in this chapter the sonic curve is a circle, and the elliptic region of the governing equation surrounds the hyperbolic region. Thus we can prescribe Dirichlet data on a suitable closed curve lying entirely in the elliptic region and obtain an elliptic–3hyperbolic boundary value problem. Eventually,we will construct such a problem and show that it possesses a weak solution. In the next chapter the sonic curve will also be a circle; but in that case the hyperbolic region of the governing equation will enclose the elliptic region, leading to a significant reduction in regularity for elliptic–hyperbolic Dirichlet problems.

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References

  1. Alías, L.J., Palmer, B: A duality result between the minimal surface equation and the maximal surface equation. An. Acad. Brasil. Ciê,nc. 73, 161–164 (2001)

    Google Scholar 

  2. Arnold, V.I.: Lectures on Partial Differential Equations. Springer-Phasis, Berlin (2004)

    Google Scholar 

  3. Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958)

    MATH  Google Scholar 

  4. Chaplygin, S.A.: On gas jets. Sci. Mem. Moscow Univ. Phys. Sec. 21, 1–121 (1902) [Translation: NACA Tech. Mem. 1063 (1944)]

    Google Scholar 

  5. Chapman, C.J.: High Speed Flow. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  6. Dinh, H., Carey, G.F.: Some results concerning approximation of regularized compressible flow. Int. J. Num. Meth. Fluids 5, 299–302 (1985)

    Article  MATH  Google Scholar 

  7. Felsen, L.B.: Evanescent waves. J. Opt. Soc. Amer. 66, 751–760 (1976)

    Article  Google Scholar 

  8. Felsen, L.B.: Complex spectra in high-frequency propagation and diffration. Proceedings of ISAP ’85, pp. 221–223

    Google Scholar 

  9. Gerlach, U.H.: Linear Mathematics in Infinite Dimensions: Signals, Boundary Value Problems, and Special Functions, Lecture 46: The Method of Steepest Descent and Stationary Phase. http://www.math.osu.edu/~gerlach/math/BVtypset/node128.html.Cited2Aug2011

  10. Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858)

    Article  MATH  Google Scholar 

  11. Hodge, W.V.D.: A Dirichlet problem for harmonic functionals with applications to analytic varieties. Proc. London Math. Soc. 36, 257–303 (1934)

    Article  Google Scholar 

  12. Iwaniec, T., Scott, C., Stroffolini, B.: Nonlinear Hodge theory on manifolds with boundary. Annali Mat. Pura Appl. 177, 37–115 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Keller, J.B.: Geometrical theory of diffraction. In: Graves, L.M. (ed.) Calculus of Variations and its Applications, Proceedings of Symposia in Applied Mathematics, vol. 8, pp. 27–52. McGraw-Hill, New York (1958)

    Google Scholar 

  14. Kravtsov, Yu.A.: A modification of the geometrical optics method [in Russian]. Radiofizika 7, 664–673 (1964)

    Google Scholar 

  15. Kreyszig, E.: On the theory of minimal surfaces. In: Rassias, Th.M. (ed.) The Problem of Plateau: A Tribute to Jesse Douglas and Tibor Radó, pp. 138–164. World Scientific, Singapore (1992)

    Chapter  Google Scholar 

  16. Ludwig, D.: Uniform asymptotic expansions at a caustic. Commun. Pure Appl. Math. 19, 215–250 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  17. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties. Contemporary Math. 283, 203–229 (1999)

    MathSciNet  Google Scholar 

  18. Magnanini, R., Talenti, G.: Approaching a partial differential equation of mixed elliptic-hyperbolic type. In: Anikonov, Yu.E., Bukhageim, A.L., Kabanikhin, S.I., Romanov, V.G. (eds.) Ill-posed and Inverse Problems, pp. 263–276. VSP, Utrecht (2002)

    Google Scholar 

  19. Magnanini, R., Talenti, G.: On complex-valued solutions to a two-dimensional eikonal equation. II. Existence theorems. SIAM J. Math. Anal. 34, 805–835 (2003)

    MATH  MathSciNet  Google Scholar 

  20. Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. III. Analysis of a Bäcklund transformation. Appl. Anal. 85, 249–276 (2006)

    MATH  MathSciNet  Google Scholar 

  21. Marini, A., Otway, T.H.: Nonlinear Hodge-Frobenius equations and the Hodge-Bäcklund transformation. Proc. R. Soc. Edinburgh, Ser. A 140, 787–819 (2010)

    Google Scholar 

  22. Morawetz, C.S.: Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation. Proc. R. Soc. London, Ser. A 236, 141–144 (1956)

    Google Scholar 

  23. Otway, T.H.: Nonlinear Hodge maps. J. Math. Phys. 41, 5745–5766 (2000)

    MATH  MathSciNet  Google Scholar 

  24. Otway, T.H.: Maps and fields with compressible density. Rend. Sem. Mat. Univ. Padova 111, 133–159 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Otway, T.H.: Variational equations on mixed Riemannian-Lorentzian metrics. J. Geom. Phys. 58, 1043–1061 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Riabouchinsky, D.: Sur l’analogie hydraulique des mouvements d’un fluide compressible. C. R. Academie des Sciences, Paris 195, 998 (1932)

    Google Scholar 

  27. Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations, Geometry and Modern Applications of Soliton Theory. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  28. Sibner, L.M., Sibner, R.J.: A nonlinear Hodge-de Rham theorem. Acta Math. 125, 57–73 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sibner, L.M., Sibner, R.J.: Nonlinear Hodge theory: Applications. Advances in Math. 31, 1–15 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sibner, L.M., Sibner, R.J., Yang, Y.: Generalized Bernstein property and gravitational strings in Born–Infeld theory. Nonlinearity 20, 1193–1213 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Talenti, G.: Some equations of non-geometrical optics. In: Lupo, D., Pagani, C.D., Ruf. B. (eds.) Nonlinear Equations: Methods, Models and Applications, pp. 257–267. Birkhäuser, Basel (2003)

    Chapter  Google Scholar 

  32. Torre, C.G.: The helically reduced wave equation as a symmetric positive system. J. Math. Phys. 44, 6223–6232 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Yang, Y.: Classical solutions in the Born-Infeld theory. Proc. R. Soc. Lond. Ser. A 456, 615–640 (2000)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Yeshiva University, New York, USA

    Thomas H. Otway

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  1. Thomas H. Otway
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Otway, T.H. (2012). Light Near a Caustic. In: The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type. Lecture Notes in Mathematics(), vol 2043. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24415-5_5

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