American Institute of Mathematical Sciences

Advanced
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), 188 (2009), 497. Google Scholar

[2]

E. Beltrami, Saggio di interpretazione della geometria non-euclidea,, Giornale di Matematiche 6 (1868), 6 (1868), 284. Google Scholar

[3]

K. O. Friedrichs, Symmetric positive linear differential equations,, Commun. Pure Appl. Math. 11 (1958), 11 (1958), 333. Google Scholar

[4]

J. Heidmann, Relativistic Cosmology, An Introduction., Springer-Verlag, (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), 77 (1951), 181. 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), 13 (1960), 427. Google Scholar

[7]

F. Lobo and P. Crawford, Time, closed timelike curves, and causality,, in The Nature of Time: Geometry, (2002), 21. Google Scholar

[8]

A. Marini and T. H. Otway, Nonlinear Hodge-Frobenius equations and the Hodge-Bäcklund transformation,, Proc. R. Soc. Edinburgh 140A (2010), 140A (2010), 787. Google Scholar

[9]

T. H. Otway, Nonlinear Hodge maps., J. Math. Phys. 41 (2000), 41 (2000), 5745. Google Scholar

[10]

T. H. Otway, Hodge equations with change of type,, Ann. Mat. Pura Appl. 181 (2002), 181 (2002), 437. Google Scholar

[11]

T. H. Otway, Harmonic fields on the projective disk and a problem in optics,, J. Math. Phys. 46 (2005), 46 (2005). Google Scholar

[12]

T. H. Otway, Variational equations on mixed Riemannian-Lorentzian metrics,, J. Geom. Phys. 58 (2008), 58 (2008), 1043. Google Scholar

[13]

T. H. Otway, The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type,, Lecture Notes in Mathematics, (2043). Google Scholar

[14]

T. H. Otway, Elliptic-Hyperbolic Partial Differential Equations: a mini-course in geometric and quasilinear methods,, Springer-Verlag, (2015). Google Scholar

[15]

L. Sarason, On weak and strong solutions of boundary value problems,, Commun. Pure Appl. Math. 15 (1962), 15 (1962), 237. Google Scholar

[16]

J. M. Stewart, Signature change, mixed problems and numerical relativity,, Class. Quantum Grav. 18 (2001), 18 (2001), 4983. Google Scholar

[17]

J. Stillwell, Sources of Hyperbolic Geometry,, Amer. Math. Soc., (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), 57 (1937), 135. Google Scholar

[19]

F. J. Tipler, Rotating cylinders and the possibility of global causality violation,, Phys. Rev. D9 (1974), D9 (1974), 2203. Google Scholar

[20]

C. G. Torre, The helically reduced wave equation as a symmetric positive system,, J. Math. Phys. 44 (2003), 44 (2003), 6223. 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), 188 (2009), 497. Google Scholar

[2]

E. Beltrami, Saggio di interpretazione della geometria non-euclidea,, Giornale di Matematiche 6 (1868), 6 (1868), 284. Google Scholar

[3]

K. O. Friedrichs, Symmetric positive linear differential equations,, Commun. Pure Appl. Math. 11 (1958), 11 (1958), 333. Google Scholar

[4]

J. Heidmann, Relativistic Cosmology, An Introduction., Springer-Verlag, (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), 77 (1951), 181. 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), 13 (1960), 427. Google Scholar

[7]

F. Lobo and P. Crawford, Time, closed timelike curves, and causality,, in The Nature of Time: Geometry, (2002), 21. Google Scholar

[8]

A. Marini and T. H. Otway, Nonlinear Hodge-Frobenius equations and the Hodge-Bäcklund transformation,, Proc. R. Soc. Edinburgh 140A (2010), 140A (2010), 787. Google Scholar

[9]

T. H. Otway, Nonlinear Hodge maps., J. Math. Phys. 41 (2000), 41 (2000), 5745. Google Scholar

[10]

T. H. Otway, Hodge equations with change of type,, Ann. Mat. Pura Appl. 181 (2002), 181 (2002), 437. Google Scholar

[11]

T. H. Otway, Harmonic fields on the projective disk and a problem in optics,, J. Math. Phys. 46 (2005), 46 (2005). Google Scholar

[12]

T. H. Otway, Variational equations on mixed Riemannian-Lorentzian metrics,, J. Geom. Phys. 58 (2008), 58 (2008), 1043. Google Scholar

[13]

T. H. Otway, The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type,, Lecture Notes in Mathematics, (2043). Google Scholar

[14]

T. H. Otway, Elliptic-Hyperbolic Partial Differential Equations: a mini-course in geometric and quasilinear methods,, Springer-Verlag, (2015). Google Scholar

[15]

L. Sarason, On weak and strong solutions of boundary value problems,, Commun. Pure Appl. Math. 15 (1962), 15 (1962), 237. Google Scholar

[16]

J. M. Stewart, Signature change, mixed problems and numerical relativity,, Class. Quantum Grav. 18 (2001), 18 (2001), 4983. Google Scholar

[17]

J. Stillwell, Sources of Hyperbolic Geometry,, Amer. Math. Soc., (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), 57 (1937), 135. Google Scholar

[19]

F. J. Tipler, Rotating cylinders and the possibility of global causality violation,, Phys. Rev. D9 (1974), D9 (1974), 2203. Google Scholar

[20]

C. G. Torre, The helically reduced wave equation as a symmetric positive system,, J. Math. Phys. 44 (2003), 44 (2003), 6223. Google Scholar

[1]

Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182
[2]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
[3]

Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43
[4]

Kyril Tintarev. Positive solutions of elliptic equations with a critical oscillatory nonlinearity. Conference Publications, 2007, 2007 (Special) : 974-981. doi: 10.3934/proc.2007.2007.974
[5]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[6]

Jun Bao, Lihe Wang, Chunqin Zhou. Positive solutions to elliptic equations in unbounded cylinder. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1389-1400. doi: 10.3934/dcdsb.2016001
[7]

Ismail Kombe. On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5167-5176. doi: 10.3934/dcds.2013.33.5167
[8]

Jeffrey R. L. Webb. Positive solutions of nonlinear equations via comparison with linear operators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5507-5519. doi: 10.3934/dcds.2013.33.5507
[9]

Yuxin Ge, Ruihua Jing, Feng Zhou. Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 751-770. doi: 10.3934/dcds.2007.17.751
[10]

Peter Poláčik. On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2657-2667. doi: 10.3934/dcds.2014.34.2657
[11]

Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039
[12]

Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151
[13]

Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943
[14]

Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
[15]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[16]

Antonio Ambrosetti, Zhi-Qiang Wang. Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 55-68. doi: 10.3934/dcds.2003.9.55
[17]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044
[18]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[19]

Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31
[20]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

 Impact Factor: 

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences