Abstract
Let (x, y) denote the variables in R 2, and consider the quasi-linear equation
where
are given smooth functions of their arguments. The equation is of order two if at least one of the coefficients A, B, C is not identically zero.
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References
Francesco Giacomo Tricorni, 1897–1978; E. Tricorni, Sulle equazioni lineari alle derivate parziali di tipo misto, Atti Accad. Naz. Lincei, Vol. 14, (1923), pp. 218–270.
See 1.3 of the Complements.
I the literature these p.d.e.’s are further classified according to the values of p and n. Namely, they are called hyperbolic if either p =1 or n = 1; otherwise they are called ultrahyperbolic.
This reduction to normal form was introduced by A.Cauchy, Mémoire sur les systèmes d’équations aux derivée partielles d’ordre quelconque, et sur leur réduction à des systémes d’équations linéaires du premier ordre, C.R. Acad. Sci. Paris, Vol. 40 (1842), pp. 131–138. Also in CEuvres Complètes d’Augustin Cauchy, Paris, Gauthiers—Villars, 1882–1974.
In the case of linear systems, the theorem was first proved by A. Cauchy, Mémoire sur les intégrales des systèmes d’équations différentielles at aux derivées partielles, et sur le developpement de ces intégrales en séries ordonnés suivant les puissances ascendentes d’un paramètre que renferment les équations proposées, C.R. Acad. Sci. Paris, Vol. 40, (1842), pp. 141–146. Also in OEuvres, cf. footnote #4. It was generalized to nonlinear systems by Sonja Kowalewski, 1850–1891: S. Kowalewski, Zur Theorie der Partiellen Differentialgleichungen, J. Reine Angew. Math. Vol. 80 (1875), pp. 1–32. A generalization is also due to G. Darboux, Sur l’existence de l’intégrale dans les équations aux derivées partielles d’ordre quelconque, C.R. Acad. Sci. Paris, Vol. 80 (1875), pp. 317–318.
The convergence of the series could be established, indirectly, by the method of the majorant. This was the original approach of A. Cauchy, followed also by S. Kowalewski and G. Darboux. For a modem account of this method, we refer to F. John [17], pp. 73–78, or to P.C. Rosenbloom, The majorant method, in Proc. of a Symposium in Pure Math., Vol. IV, Amer. Math. Soc., Providence R.I. (1961). The convergence of the series, can also be established by a direct estimation of all the derivatives of u. This is the method we present here. This approach is due to P.D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math. Vol. 4 (1953), pp. 231–258. It has been further elaborated and extended by A. Friedman, A new proof and generalizations of the Cauchy—Kowalewski theorem, Trans. Amer. Math. Soc. #98, (1961), pp.1–20. It has also been extended by Shimbrot and Welland, to an infinite-dimensional setting: M. Shimbrot and R.E. Welland, The Cauchy—Kowalewski theorem, J. Math. Anal. and Appl. Vol 25 #3, Sept. 1976, pp. 757–772.
See 6.1 of the Complements.
See 6.3 of the Complements.
See 6.4 of the Complements.
Gottfried Wilhelm Leibniz, 1646–1716; see 8.1 of the Complements.
See 9.1 of the Complements.
See 9.2 of the Complements.
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DiBenedetto, E. (1995). Quasi-Linear Equations and the Cauchy—Kowalewski Theorem. In: Partial Differential Equations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-2840-5_2
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DOI: https://doi.org/10.1007/978-1-4899-2840-5_2
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