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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

Antonella Marini 1, and  Thomas H. Otway 2,

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.
Keywords: extended projective disc, Elliptic-hyperbolic equations, symmetric positive operators..
Mathematics Subject Classification: Primary: 35M32; Secondary: 35Q75, 58J3.
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

show all references

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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