Abstract
A second-order quasi-linear partial differential equation of mixed elliptic-hyperbolic type in two independent variables, which mimics one introduced by A. Busemann in gas dynamics, and arises in the study of Minkowski spaces and in other theories, is considered.
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Abramowitz M., Stegun I.A.: Handbook of mathematical functions. Dover Publications, New York (1972)
Alías L.J., Chaves R.M.B., Mira P.: Björling problem for maximal surfaces in Lorentz-Minkowski space. Mat. Proc. Cambridge Philos. Soc. 134, 289–316 (2003)
Alías L.J., Palmer B.: A duality result between the minimal surface equation and the maximal surface equation. An. Acad. Bras. Cienc. 73, 161–164 (2001)
R. L. Anderson, N. H. Ibragimov, Lie-Bäcklund transformations in applications. SIAM, 1979.
Arnold V.I.: Mathematical methods of classical mechanics. Springer-Verlag, New York (1978)
G. Aronsson, A stream function technique for the p-harmonic equation in the plane. Department of Mathematics, University of Luleå, 1986-3.
Aronsson G.: Representation of a p-harmonic function near a critical point in the plane. Manuscripta Mathematica 66, 73– (1986)
Aronsson G.: On certain p-harmonic functions in the plane. Manuscripta Math. 61, 79–101 (1988)
Aronsson G., Lindqvist P.: On p-harmonic functions in the plane and their stream functions. J. Diff. Equations 74, 157– (1988)
R. Bartnik, The existence of maximal surfaces. Miniconference on operator theory and partial differential equations (Canberra, 1983), 47–51, Proc. CentreMath. Anal. Austral. Nat. Univ., 5, Austral. Nat. Univ., Canberra, 1984.
Bartnik R.: Existence of maximal surfaces in asymptotically flat spacetimes. Comm. Math. Phys. 94, 155–175 (1984)
R. Bartnik, Maximal surfaces and general relativity. Miniconference on geometry and partial differential equations 2 (Canberra, 1986), 24–49, Proc. Centre Math. Anal. Austral. Nat. Univ., 12, Austral. Nat. Univ., Canberra, 1987.
Bartnik R.: Regularity of variational maximal surfaces. Acta Math. 166, 145–181 (1988)
Bartnik R., Chrusciel P.T., Murchadha O.: On maximal surfaces in asymptotically flat space-time. Comm. Math. Phys. 130, 95–109 (1990)
Bartnik R., Simon L.: Spacelike hypersurfaces with prescribed boundary values and maen curvature. Comm. Math. Phys. 87, 131– (1982)
L. Bers, Mathematical aspects of subsonic and transonic gas dynamics. Surveys in Applied Mathematics 3, John Wiley and Sons, 1958.
A. V. Bitsadze (A. B. Бицадзе), On some problems of mixed type (Russian). Dokl. Akad. Nauk SSSR 70 (1950), 561–564.
A. V. Bitsadze (A. B. Бицадзе), On the problem of equations of mixed type (Russian). Trudy Mat. Inst. Steklov 41, Izdat. Akad. Nauk SSSR 1953.
A. V. Bitsadze (A. B. Бицадзе), Incorrectness of Dirichlet’s problem for the mixed type of equations in mixed regions (Russian). Dokl. Akad. Nauk SSSR 122 (1958), 167–170.
A. V. Bitsadze (A. B. Бицадзе), Differential equations of mixed type. Macmillan Co., New York, 1964.
Busemann A.: Die achsensymmetrische kegelige Überschallströmung. Luftfahrtforshung 19, 137–144 (1942)
Busemann A.: Infinitesimale kegelige Überschallströmung. Schriften der Deutschen Akademie für Luftfahrtforshung 7, 105–121 (1943)
Calabi E.: Examples of Bernstein problems for some non-linear equations. Proc. Symposia Pure Math. 15, 223–230 (1970)
Cannon J.R.: A Dirichlet problem for an equation of mixed type with a discontinuous coefficient. Ann. Mat. Pura Appl. 62, 371–377 (1963)
Chen W., Su N.: self-conjugate maximal surfaces in L 3. Northeast. Math. J. 14, 9–16 (1998)
Cheng S.Y., Yau S.T.: Maximal spacelike surfaces in Lorentz-Minkowski spaces. Ann. Math. 104, 407–419 (1976)
Cole J.D.: On a quasilinear parabolic equation occurring in aerodinamics. Quart. Appl. Math. 9, 225–236 (1951)
R. Courant, Dirichlet’s principle, conformal mapping, and minimal surfaces. Dover Publications, 1950.
R. Courant, D. Hilbert, Methods of Mathematical Physics, volume II. Interscience Publishers, 1962.
U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal surfaces. Springer-Verlag, 2010.
Dodd R.K., Eilbeck J.C., Gibbon J.D., Morris H.C.: Solitons and nonlinear wave equations. Academic Press, London (1982)
Ecker K.: On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetimes. J. Austral. Math. Soc. Ser. A 55, 41–59 (1993)
Ecker K., Huisken G.: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Comm. Math. Phys. 135, 595–613 (1991)
Ecker K.: Interior estimates and longtime solutions for mean curvature flow of non compact spacelike hypersurfaces in Minkowski space. J. Differential Geometry 45, 481–498 (1997)
Einzinger P., Raz S.: On the asymptotic theory of inhomogeneous wave tracking. Radio Science 15, 763–771 (1980)
L. C. Evans, Partial Differential Equations. Graduate Studies in Math. 19, Amer. Math. Soc., 1998.
Felsen L.B.: Evanescent waves. J. Opt. Soc. Am. 66, 751–760 (1976)
Paul Germain, La théorie générale des mouvements coniques et ses applications à l’aérodynamique supersonique. Office National d’Etudes et de Recherches Aeronautiques, publication no. 34, 1949.
Paul Germain: La théorie des mouvements homognèes et son application au calcul de certain ailes delta en régime supersonique. Recherche Aéronautique 7, 3–16 (1949)
Gu C.H.: The extremal surfaces in the 3-dimensional Minkowski space. Acta Math. Sinica (N.S.) 1(2), 173–180 (1985)
Gu C.H.: A class of boundary problems for estremal surfaces of mixed type in Minkowski 3-space. J. Reine Angew. Math. 385, 195–202 (1988)
Hochstadt H.: On a partial differential equation of mixed type with discontinuous coefficients. Arch. Rat. Mech. Analysis 10, 267–272 (1962)
Hopf E.: The partial differential equation u t + uu x = µ u xx . Comm. Pure Appl. Math. 3, 201–230 (1950)
K. Huang, Statistical Mechanics. John Wiley and Sons, 1987.
Klyachin V.A.: Investigation of solutions of an equation of surfaces with zero mean curvature of mixed type in Minkowski space. Dokl. Akad. Nauk 384, 587–589 (2002)
Kobayashi O.: Maximal surfaces in the 3-dimensional Minkowski space. Tokyo J. Math. 6, 297–309 (1983)
M. A. Lavrentiev (M. A. Лаврентиев), A. V. Bitsadze (A. B. Бицадзе), On the problem of equations of mixed type (Russian). Dokl. Akad. Nauk SSSR 70 (1950), 373–376.
M. A. Lavrentiev (M. A. Лаврентиев), B. V. Chabat (Б. B. Шабат), Effets hydrodynamiques et modèles mathématiques. Editions MIR, Moscow, 1980 (French translation of the 1977 Russian edition).
J. D. Lawrence, A catalog of special plane curves. Dover Publications, 1972.
Li J.: Stationary surfaces in Minkowski space. I. A representation formula. Pacific J. Math. 158, 353–363 (1993)
H. W. Liepmann, A. Roshko, Elements of Gasdynamics. Dover Publications, 1957 and 2001.
Lindblad H.: A remark on global existence for small initial data of the minimal surface equation in Minkowskian spacetime. Proc. Amer.Math. Soc. 321(4), 1095–1102 (2003)
Liu H.L.: Minimal time-like surfaces in three-dimensional Minkowski space. J. Northeast Univ. Tech. 12, 308–310 (1991)
Liu H.L.: The general Weierstrass formula for surfaces in 3-dimensional Minkowski space. Acta Math. Sinica 38, 191–199 (1995)
Magnanini R., Talenti G.: On complex-valued solutions to a 2D eikonal equation. Part One: qualitative properties. Contemporary Math. 283, 203–229 (1999)
Magnanini R., Talenti G.: On complex-valued solutions to a 2D eikonal equation. Part Two: existence theorems. SIAM J. Math. Anal. 34, 805–835 (2002)
R. Magnanini, G. Talenti, Approaching a partial differential equation of mixed elliptichyperbolic type. Pages 263–276 in Ill-posed and Inverse Problems (S.I. Kabanikin and V.G. Romanov Editors), VSP, Netherlands (2002).
Magnanini R., Talenti G.: On complex-valued solutions to a 2D eikonal equation. Part Three: analysis of a Bäcklund transformation. Appl. Anal. 85(1-3), 249–276 (2006)
Magnanini R., Talenti G.: On complex-valued 2D eikonals. Part Four: continuation past a caustic. Milan J. Math. 77, 1–66 (2009)
Mazet L.: A uniqueness result for maximal surfaces in Minkowski 3-space. C. R. Acad. Sci. Paris Ser. I 344, 785–790 (2007)
J. C. Nitsche, Lectures on minimal surfaces. Cambridge Univ. Press, 1989.
R. Osserman, A survey of minimal surfaces. Dover Publications, 2002.
Rockafellar R.T.: Convex analysis. Princeton University Press, Princeton (NJ) (1970)
C. Rogers,W. K. Schief, Bäcklund and Darboux transformations. Cambridge University Press, 2002.
C. Rogers,W. K. Schief, M. E. Johnston, Bäcklund and his work: applications in soliton theory. Pages 16–55 in Geometric approaches to differential equations, P.J. Vassiliou and I.G. Lisle Editors, Cambridge University Press, 2000.
C. Rogers,W. F. Shadwick, Bäcklund transformations and their applications. Academic Press, 1982.
Romero A.: Simple proof of Calabi-Bernstein theorem on maximal surfaces. Proc. Amer. Math. Soc. 124, 1315–1317 (1996)
J. Serrin, Mathematical principles of classical fluid mechanics. Handbuch der Physik, volume 8 (1959), pages 125–263.
M. M. Smirnov (M. M. Смирнов), Equations of mixed type. Translations of Math. Monographs no. 51, Amer. Math. Soc., 1978.
Umehara M., Yamada K.: Maximal surfaces with singularities in Minkowski space. Hokkaido Math. J. 35, 13– (2006)
I. Van de Woesijne, Minimal surfaces of the 3-dimensional Minkowski space. Geometry and topology of submanifolds II (Avignon, 1988), 344–369, World Sci. Publ., 1990.
D. Zwillinger, Handbook of differential equations. Academic Press, 1997.
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Talenti, G. Thoughts on the Busemann equation. Milan J. Math. 79, 145–180 (2011). https://doi.org/10.1007/s00032-011-0153-8
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DOI: https://doi.org/10.1007/s00032-011-0153-8
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Keywords
- Second-order quasi-linear partial differential equations of mixed type
- first-order non-linear partial differential systems
- Bäcklund transformations
- complex-valued eikonals
- minimal surfaces
- space-like maximal surfaces in Minkowski space
- Legendre transformation
- Lavrentiev-Bitsadze equation
- D’Alembert equation
- initial value problems