Geodesic
Hyperbolic formulas in elliptic Сauchy problems
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 3 (2010) no. 4, pp. 419-432.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of it's boundary which is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of Hans Lewy (1927).
Keywords: Cauchy problem, wave equation
Mots-clés : Laplace equation, Carleman formulas. 
@article{JSFU_2010_3_4_a0,
     author = {Dmitry P. Fedchenko and Nikolai Tarkhanov},
     title = {Hyperbolic formulas in elliptic {{\CYRS}auchy} problems},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {419--432},
     publisher = {mathdoc},
     volume = {3},
     number = {4},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2010_3_4_a0/}
}
TY  - JOUR
AU  - Dmitry P. Fedchenko
AU  - Nikolai Tarkhanov
TI  - Hyperbolic formulas in elliptic Сauchy problems
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2010
SP  - 419
EP  - 432
VL  - 3
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2010_3_4_a0/
LA  - en
ID  - JSFU_2010_3_4_a0
ER  - 
%0 Journal Article
%A Dmitry P. Fedchenko
%A Nikolai Tarkhanov
%T Hyperbolic formulas in elliptic Сauchy problems
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2010
%P 419-432
%V 3
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2010_3_4_a0/
%G en
%F JSFU_2010_3_4_a0
Dmitry P. Fedchenko; Nikolai Tarkhanov. Hyperbolic formulas in elliptic Сauchy problems. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 3 (2010) no. 4, pp. 419-432. http://geodesic.mathdoc.fr/item/JSFU_2010_3_4_a0/

[1] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven–London, 1923 | Zbl

[2] C. Morrey, “On the analyticity of the solutions of analytic nonlinear elliptic systems of partial differential equations. II. Analyticity at the boundary”, Amer. J. Math., 80 (1958), 219–237 | DOI | MR | Zbl

[3] P. Lax, “Asymptotic solutions of oscillatory initial value problems”, Duke Math. J., 24 (1957), 627–646 | DOI | MR | Zbl

[4] S. Mizohata, “Some remarks on the Cauchy problem”, J. Math. Kyoto Univ., 1 (1961), 109–127 | MR | Zbl

[5] T. Nishitani, “A note on the local solvability of the Cauchy problem”, J. Math. Kyoto Univ., 24 (1984), 281–284 | MR | Zbl

[6] S. Wakabayashi, “The Lax–Mizohata theorem for nonlinear Cauchy problems”, Comm. in Part. Diff. Equ., 26 (2001), 1367–1384 | DOI | MR | Zbl

[7] J. Hounie, J. R. Filho dos Santos, “Well-posed Cauchy problems for complex nonlinear equations must be semilinear”, Math. Ann., 294 (1992), 439–447 | DOI | MR | Zbl

[8] Guy Métivier, Remarks on the well-posedness of the nonlinear Cauchy problem, 14 Nov 2006, 20 pp., arXiv: math/0611441v1[math.AP] | MR

[9] H. Lewy, “Neuer Beweis des analytischen Charakters der Lösungen elliptischer Differentialgleichungen”, Math. Ann., 101, 1927 | MR

[10] Sh. Yarmukhamedov, “On the Cauchy problem for Laplace's equation”, Math. Notes, 18:1 (1975), 615–618 | MR | Zbl

[11] A. A. Shlapunov, “On the Cauchy problem for the Laplace equation”, Sibirsk. Mat. Zh., 33:3 (1992), 205–215 | MR | Zbl

[12] J. Leray, “Problème de Cauchy, I”, Bull. Soc. Math. France, 85 (1957), 389–439 ; “II”, 86 (1958), 75–96 ; “III”, 87 (1959), 81–180 ; “IV”, 90 (1962), 39–156 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[13] J. Leray, “The functional transformations required by the theory of partial differential equations”, SIAM Review, 5 (1963), 321–334 | DOI | MR | Zbl

[14] T. Carleman, Les fonctions quasianalytiques, Gauthier-Villars, Paris, 1926 | Zbl

[15] N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995 | MR | Zbl

[16] A. Krylov, “A Cauchy problem for Laplace's equation in the complex domain”, Dokl. Akad. Nauk SSSR, 188:4 (1969), 748–751 | MR | Zbl

[17] L. Aizenberg, Carleman Formulas in Complex Analysis, Kluwer Academic Publishers, Dordrecht, NL, 1993 | MR

[18] R. Courant, D. Hilbert, Methoden der mathematischen Physik, v. II, Auflage, Springer-Verlag, Berlin et al., 1968 | MR | Zbl