Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge–Bäcklund transformation, underlying symmetries among superficially different forms of the equations.
@article{AIHPC_2014__31_2_339_0, author = {Marini, Antonella and Otway, Thomas H.}, title = {Duality methods for a class of quasilinear systems}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {31}, year = {2014}, pages = {339-348}, doi = {10.1016/j.anihpc.2013.03.007}, mrnumber = {3181673}, zbl = {1300.35047}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_2_339_0} }
Marini, Antonella; Otway, Thomas H. Duality methods for a class of quasilinear systems. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 339-348. doi : 10.1016/j.anihpc.2013.03.007. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_2_339_0/
[1] Advances in differential forms and the A-harmonic equations, Math. Comput. Modelling 37 (2003), 1393-1426 | MR 1996046 | Zbl 1051.58001
, ,[2] Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces, J. Geom. Physics 59 (2009), 620-631 | MR 2518991 | Zbl 1173.53025
, ,[3] A duality result between the minimal surface equation and the maximal surface equation, An. Acad. Brasil. Ciênc 73 (2001), 161-164 | MR 1833778 | Zbl 0999.53007
, ,[4] Applied Exterior Calculus, Wiley, New York (1985) | MR 816136 | Zbl 0386.73002
,[5] Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl. 177 (1999), 37-115 | MR 1747627 | Zbl 0963.58003
, , ,[6] Extensions of the duality between minimal surfaces and maximal surfaces, Geom. Dedicata 151 (2011), 373-386 | MR 2780757 | Zbl 1211.53010
,[7] On complex-valued solutions to a 2D eikonal equation. Part one: qualitative properties, Contemporary Math. 283 (1999), 203-229 | MR 1724665 | Zbl 0940.35100
, ,[8] Approaching a partial differential equation of mixed elliptic–hyperbolic type, , , , (ed.), Ill-posed and Inverse Problems, VSP (2002), 263-276
, ,[9] Nonlinear Hodge–Frobenius equations and the Hodge–Bäcklund transformation, Proc. R. Soc. Edinburgh A 140 (2010), 787-819 | MR 2672070 | Zbl 1209.58003
, ,[10] Constructing completely integrable fields by the method of generalized streamlines, arXiv:1205.7028 [math.AP] | Zbl 1305.35116
, ,[11] Decomposition and their application to nonlinear electro- and magnetostatic boundary value problems, , (ed.), Partial Differential Equations and Calculus of Variations, Lecture Notes in Mathematics vol. 1357, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1988) | Zbl 0684.35084
, ,[12] Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin (1966) | MR 202511 | Zbl 0142.38701
,[13] Nonlinear Hodge maps, J. Math. Phys. 41 (2000), 5745-5766 | MR 1773064 | Zbl 0974.58018
,[14] Maps and fields with compressible density, Rend. Sem. Mat. Univ. Padova 111 (2004), 133-159 | Numdam | MR 2076737 | Zbl 1121.76056
,[15] The Dirichlet Problem for Elliptic–Hyperbolic Equations of Keldysh Type, Lecture Notes in Mathematics vol. 2043, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (2012) | MR 2933771 | Zbl 1246.35006
,[16] Hodge Decomposition: A Method for Solving Boundary Value Problems, Lecture Notes in Mathematics vol. 1607, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1995) | MR 1367287 | Zbl 0828.58002
,[17] A nonlinear Hodge–de Rham theorem, Acta Math. 125 (1970), 57-73 | MR 281231 | Zbl 0216.45703
, ,[18] Nonlinear Hodge theory: Applications, Advances in Math. 31 (1979), 1-15 | MR 521463 | Zbl 0408.58032
, ,[19] Generalized Bernstein property and gravitational strings in Born–Infeld theory, Nonlinearity 20 (2007), 1193-1213 | MR 2312389 | Zbl 1117.58018
, , ,[20] Classical solutions in the Born–Infeld theory, Proc. R. Soc. Lond. Ser. A 456 (2000), 615-640 | MR 1808753 | Zbl 1122.78301
,