# American Institute of Mathematical Sciences

2015, 2015(special): 801-808. doi: 10.3934/proc.2015.0801

## Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric

 1 Dipartimento di Matematica, Università di L'Aquila, 67100 L'Aquila, Italy 2 Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, United States

Received  September 2014 Revised  January 2015 Published  November 2015

A first-order elliptic-hyperbolic system in extended projective space is shown to possess strong solutions to a natural class of Guderley--Morawetz--Keldysh problems on a typical domain.
Citation: Antonella Marini, Thomas H. Otway. Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric. Conference Publications, 2015, 2015 (special) : 801-808. doi: 10.3934/proc.2015.0801
##### References:
 [1] J. Barros-Neto and F. Cardoso, Gellerstedt and Laplace-Beltrami operators relative to a mixed signature metric, Ann. Mat. Pura Appl. 188 (2009), 497-515. Google Scholar [2] E. Beltrami, Saggio di interpretazione della geometria non-euclidea, Giornale di Matematiche 6 (1868), 284-312. Google Scholar [3] K. O. Friedrichs, Symmetric positive linear differential equations, Commun. Pure Appl. Math. 11 (1958), 333-418. Google Scholar [4] J. Heidmann, Relativistic Cosmology, An Introduction. Springer-Verlag, Berlin-Heidelberg-New York (1980). Google Scholar [5] M. V. Keldysh, On certain classes of elliptic equations with singularity on the boundary of the domain (Russian]), Dokl. Akad. Nauk SSSR 77 (1951), 181-183. Google Scholar [6] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Commun. Pure Appl. Math. 13 (1960), 427-455. Google Scholar [7] F. Lobo and P. Crawford, Time, closed timelike curves, and causality, in The Nature of Time: Geometry, Physics and Perception (NATO ARW), Proceedings of a conference held 21-24 May, 2002 at Tatranska Lomnica, Slovak Republic (eds. Rosolino Buccheri, Metod Saniga, and William Mark Stuckey). NATO Science Series II: Mathematics, Physics and Chemistry - Volume 95. Dordrecht/Boston/London: Kluwer Academic Publishers, 2003. Google Scholar [8] A. Marini and T. H. Otway, Nonlinear Hodge-Frobenius equations and the Hodge-Bäcklund transformation, Proc. R. Soc. Edinburgh 140A (2010), 787-819. Google Scholar [9] T. H. Otway, Nonlinear Hodge maps. J. Math. Phys. 41 (2000), 5745-5766. Google Scholar [10] T. H. Otway, Hodge equations with change of type, Ann. Mat. Pura Appl. 181 (2002), 437-452. Google Scholar [11] T. H. Otway, Harmonic fields on the projective disk and a problem in optics, J. Math. Phys. 46 (2005), 113501. (Erratum: J. Math. Phys. 48 (2007), 079901.) Google Scholar [12] T. H. Otway, Variational equations on mixed Riemannian-Lorentzian metrics, J. Geom. Phys. 58 (2008), 1043-1061. Google Scholar [13] T. H. Otway, The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type, Lecture Notes in Mathematics, Vol. 2043, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 2012. Google Scholar [14] T. H. Otway, Elliptic-Hyperbolic Partial Differential Equations: a mini-course in geometric and quasilinear methods, Springer-Verlag, London, 2015. Google Scholar [15] L. Sarason, On weak and strong solutions of boundary value problems, Commun. Pure Appl. Math. 15 (1962), 237-288. Google Scholar [16] J. M. Stewart, Signature change, mixed problems and numerical relativity, Class. Quantum Grav. 18 (2001), 4983-4995. Google Scholar [17] J. Stillwell, Sources of Hyperbolic Geometry, Amer. Math. Soc., Providence, 1996. Google Scholar [18] W. J. van Stockum, The gravitational field of a distribution of particles rotating about an axis of symmetry, Proc. R. Soc. Edinburgh 57 (1937), 135-154. Google Scholar [19] F. J. Tipler, Rotating cylinders and the possibility of global causality violation, Phys. Rev. D9 (1974), 2203-2206. Google Scholar [20] C. G. Torre, The helically reduced wave equation as a symmetric positive system, J. Math. Phys. 44 (2003), 6223-6232. Google Scholar

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##### References:
 [1] J. Barros-Neto and F. Cardoso, Gellerstedt and Laplace-Beltrami operators relative to a mixed signature metric, Ann. Mat. Pura Appl. 188 (2009), 497-515. Google Scholar [2] E. Beltrami, Saggio di interpretazione della geometria non-euclidea, Giornale di Matematiche 6 (1868), 284-312. Google Scholar [3] K. O. Friedrichs, Symmetric positive linear differential equations, Commun. Pure Appl. Math. 11 (1958), 333-418. Google Scholar [4] J. Heidmann, Relativistic Cosmology, An Introduction. Springer-Verlag, Berlin-Heidelberg-New York (1980). Google Scholar [5] M. V. Keldysh, On certain classes of elliptic equations with singularity on the boundary of the domain (Russian]), Dokl. Akad. Nauk SSSR 77 (1951), 181-183. Google Scholar [6] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Commun. Pure Appl. Math. 13 (1960), 427-455. Google Scholar [7] F. Lobo and P. Crawford, Time, closed timelike curves, and causality, in The Nature of Time: Geometry, Physics and Perception (NATO ARW), Proceedings of a conference held 21-24 May, 2002 at Tatranska Lomnica, Slovak Republic (eds. Rosolino Buccheri, Metod Saniga, and William Mark Stuckey). NATO Science Series II: Mathematics, Physics and Chemistry - Volume 95. Dordrecht/Boston/London: Kluwer Academic Publishers, 2003. Google Scholar [8] A. Marini and T. H. Otway, Nonlinear Hodge-Frobenius equations and the Hodge-Bäcklund transformation, Proc. R. Soc. Edinburgh 140A (2010), 787-819. Google Scholar [9] T. H. Otway, Nonlinear Hodge maps. J. Math. Phys. 41 (2000), 5745-5766. Google Scholar [10] T. H. Otway, Hodge equations with change of type, Ann. Mat. Pura Appl. 181 (2002), 437-452. Google Scholar [11] T. H. Otway, Harmonic fields on the projective disk and a problem in optics, J. Math. Phys. 46 (2005), 113501. (Erratum: J. Math. Phys. 48 (2007), 079901.) Google Scholar [12] T. H. Otway, Variational equations on mixed Riemannian-Lorentzian metrics, J. Geom. Phys. 58 (2008), 1043-1061. Google Scholar [13] T. H. Otway, The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type, Lecture Notes in Mathematics, Vol. 2043, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 2012. Google Scholar [14] T. H. Otway, Elliptic-Hyperbolic Partial Differential Equations: a mini-course in geometric and quasilinear methods, Springer-Verlag, London, 2015. Google Scholar [15] L. Sarason, On weak and strong solutions of boundary value problems, Commun. Pure Appl. Math. 15 (1962), 237-288. Google Scholar [16] J. M. Stewart, Signature change, mixed problems and numerical relativity, Class. Quantum Grav. 18 (2001), 4983-4995. Google Scholar [17] J. Stillwell, Sources of Hyperbolic Geometry, Amer. Math. Soc., Providence, 1996. Google Scholar [18] W. J. van Stockum, The gravitational field of a distribution of particles rotating about an axis of symmetry, Proc. R. Soc. Edinburgh 57 (1937), 135-154. Google Scholar [19] F. J. Tipler, Rotating cylinders and the possibility of global causality violation, Phys. Rev. D9 (1974), 2203-2206. Google Scholar [20] C. G. Torre, The helically reduced wave equation as a symmetric positive system, J. Math. Phys. 44 (2003), 6223-6232. Google Scholar
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