Abstract
In the introductory chapters to most plasma physics texts (e.g., [43,45]), an idealized model is presented in which the plasma ion and electron temperatures – rather than being millions of degrees – are set to absolute zero. This is done to reduce the mathematics to its simplest possible form, but even in this case a rigorous description of the plasma is problematic: the classical Dirichlet problem, which is the physically natural boundary value problem in this context, turns out to be ill-posed on typical domains.
Keywords
- Dirichlet Problem
- Hyperbolic Equation
- Isometric Embedding
- Beltrami Equation
- Inhomogeneous Boundary Condition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator. Duke Math. J. 98, 465–483 (1999)
Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator II. Duke Math. J. 111, 561–584 (2002)
Barros-Neto, J., Gelfand, I.M.: Fundamental solutions for the Tricomi operator III, Duke Math. J. 128, 119–140 (2005)
Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958)
Čanić, S., Keyfitz, B.: A smooth solution for a Keldysh type equation. Commun. Partial Diff. Equations 21, 319–340 (1996)
Chapman, C.J.: High Speed Flow. Cambridge University Press, Cambridge (2000)
Chen, G-Q., Slemrod, M., Wang, D.: Isometric immersions and compensated compactness. Commun. Math. Phys. 294, 411–437 (2010)
Cibrario, M.: Sulla riduzione a forma canonica delle equazioni lineari alle derivate parziali di secondo ordine di tipo misto. Rendiconti del R. Insituto Lombardo 65 (1932)
Cinquini-Cibrario, M.: Review of Rassias, John M. (GR-UATH) Lecture notes on mixed type partial differential equations, World Scientific Publishing Co., Teaneck, 1990. In: Mathematical Reviews, MR1082555 (91m:35162)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Mechanics. Springer, Berlin (2005)
Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Diff. Equations 7, 77–116 (1982)
Frankl’, F.L.: Problems of Chaplygin for mixed sub- and supersonic flows [in Russian]. Izv. Akad. Nauk SSSR, ser. mat. 9(2), 121–143 (1945)
Gramchev, T.V.: An application of the analytic microlocal analysis to a class of differential operators of mixed type. Math. Nachr. 121, 41–51 (1985)
Groothuizen, R.J.P.: Mixed Elliptic-Hyperbolic Partial Differential Operators: A Case Study in Fourier Integral Operators. CWI Tract, vol. 16, Centrum voor Wiskunde en Informatica, Amsterdam (1985)
Gu, C.: On partial differential equations of mixed type in n independent variables. Commun. Pure Appl. Math. 34, 333–345 (1981)
Han, Q.: Local solutions to a class of Monge-Ampère equations of mixed type. Duke Math. J. 136 421–473 (2007)
Han, Q., Hong, J.-X.: Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence (2006)
Han, Q., Khuri, M.: On the local isometric embedding in R 3 of surfaces with Gaussian curvature of mixed sign. Commun. Anal. Geom. 18, 649–704 (2010)
Hersh, R.: How to classify differential polynomials. Amer. Math. Monthly 80, 641–654 (1973)
Jang, J., Masmoudi, N.: Well-posedness for compressible Euler equations with physical vacuum singularity. Commun. Pure Appl. Math. 62, 1327–1385 (2009)
Keldysh, M.V.: On certain classes of elliptic equations with singularity on the boundary of the domain [in Russian]. Dokl. Akad. Nauk SSSR 77, 181–183 (1951)
Khuri, M.A.: The local isometric embedding in R 3 of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve. J. Differential Geom. 76, 249–291 (2007)
Khuri, M.A.: Boundary value problems for mixed type equations and applications. J. Nonlinear Anal. Ser. A: TMA (to appear); arXiv:1106.4000v1
Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)
Lin, C.S.: The local isometric embedding in R 3 of a 2-dimensional Riemanian manifolds with Gaussian curvature changing sign cleanly. Commun. Pure Appl. Math. 39, 867–887 (1986)
Lupo, D., Morawetz, C.S., Payne, K.R.: On closed boundary value problems for equations of mixed elliptic-hyperbolic type. Commun. Pure Appl. Math. 60, 1319–1348 (2007)
Lupo, D., Morawetz, C.S., Payne, K.R.: Erratum: “On closed boundary value problems for equations of mixed elliptic-hyperbolic type,” [Commun. Pure Appl. Math. 60, 1319–1348 (2007)]. Commun. Pure Appl. Math. 61, 594 (2008)
Magnanini, R., Talenti, G.: On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties. Contemp. Math. 283, 203–229 (1999)
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)
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)
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)
Magnanini, R., Talenti, G.: On complex-valued 2D eikonals. IV. Continuation past a caustic. Milan J. Math. 77, 1–66 (2009)
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)
Morawetz, C.S., Stevens, D.C., Weitzner, H.: A numerical experiment on a second-order partial differential equation of mixed type. Commun. Pure Appl. Math. 44, 1091–1106 (1991)
Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)
Otway, T.H.: Hodge equations with change of type. Ann. Mat. Pura Appl. 181, 437–452 (2002)
Otway, T.H.: Harmonic fields on the extended projective disc and a problem in optics. J. Math. Phys. 46, 113501 (2005)
Otway, T.H.: Erratum: Harmonic fields on the extended projective disc and a problem in optics [J. Math. Phys. 46, 113501 (2005)]. J. Math. Phys. 48, 079901 (2007)
Otway, T.H.: Energy inequalities for a model of wave propagation in cold plasma. Publ. Mat. 52, 195–234 (2008)
Parker, P.E.: Geometry of bicharacteristics. In: Advances in Differential Geometry and General Relativity, Contemp. Math., vol. 359, pp. 31–40. American Mathematical Society, Providence (2004)
Payne, K.R.: Propagation of singularities for solutions to the Dirichlet problem for equations of Tricomi type, Rend. Sem. Mat. Univ. Pol. Torino 54, 115–137 (1996)
Payne, K.R.: Interior regularity of the Dirichlet problem for the Tricomi equation. J. Mat. Anal. Appl. 199, 271–292 (1996)
Payne, K.R.: Solvability theorems for linear equations of Tricomi type. J. Mat. Anal. Appl. 215, 262–273 (1997)
Rassias, J.M.: Lecture notes on Mixed Type Partial Differential Equations. World Scientific, Teaneck (1990)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, Berlin (1983)
Stix, T.H.: The Theory of Plasma Waves. McGraw-Hill, New York (1962)
Stoker, J.J.: Water Waves. Interscience, New York (1987)
Stokes, G.G.: Mathematical and Physical Papers, vol. 5, Appendix. Cambridge University Press, Cambridge (1880–1905)
Swanson, D.G.: Plasma Waves. Institute of Physics, Bristol (2003)
Taylor, M.E.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)
Zheng, Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems. Birkhauser, Boston (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Otway, T.H. (2012). Introduction. 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_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-24415-5_1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24414-8
Online ISBN: 978-3-642-24415-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
