Geodesic
Negative Sobolev Spaces in the Cauchy Problem for the Cauchy--Riemann Operator
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 2 (2009) no. 1, pp. 17-30.

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Let $D$ be a bounded domain in $\mathbb C^n$ ($n\ge1$) with a smooth boundary $\partial D$. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for the Cauchy–Riemann operator $\overline\partial$ in $D$. In particular, we describe traces of the corresponding Sobolev functions on $\partial D$ and give an adequate formulation of the problem. Then we prove the uniqueness theorem for the problem, describe its necessary and sufficient solvability conditions and produce a formula for its exact solution.
Keywords: negative Sobolev spaces, ill-posed Cauchy problem. 
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Ivan V. Shestakov; Alexander A. Shlapunov. Negative Sobolev Spaces in the Cauchy Problem for the Cauchy--Riemann Operator. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 2 (2009) no. 1, pp. 17-30. http://geodesic.mathdoc.fr/item/JSFU_2009_2_1_a1/

[1] S. L. Sobolev, Equations of Mathematical Physics, Fizmatgiz, Moscow, 1954 (Russian) | MR

[2] O. A. Ladyzhenskaya, Boundary Value Problems for Equations of Mathematical Physics, Nauka, Moscow, 1973 (Russian) | MR

[3] M. Schechter, “Negative norms and boundary problems”, Ann. Math., 72:3 (1960), 581–593 | DOI | MR | Zbl

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

[5] E. J. Straube, “Harmonic and analytic functions admitting a distribution boundary value”, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11:4 (1984), 559–591 | MR | Zbl

[6] J. Hadamard, Le problème de Cauchy et les equations aux derivées partielles linéares hyperboliques, Gauthier-Villars, Paris, 1932 | Zbl

[7] L. A. Aizenberg, Carleman's Formulas in Complex Analysis. First Applications, Kluwer, Dordreht–Boston–London, 1993

[8] T. Carleman, Les Fonctions Quasianalytiques, Gauthier-Villars, Paris, 1926 | Zbl

[9] L. A. Aizenberg, A. M. Kytmanov, “On possibility of holomorphic extension to a domain of functions, given on a part of its boundary”, Mat. USSR–Sb., 72:2 (1992), 467–483 | DOI | MR

[10] A. A. Shlapunov, N. N. Tarkhanov, “On the Cauchy problem for holomorphic functions of the Lebesque class $L^2$ in a domain”, Sib. Math. J., 33:5 (1992), 914–922 | DOI | MR | Zbl

[11] D. P. Fedchenko, A. A. Shlapunov, “On the Cauchy problem for multi-dimensional Cauchy–Riemann operator in Lebesque $L^2$ in a domain”, Mat. Sb., 199:11 (2008), 141–160 (Russian)

[12] L. Hörmander, Notions of Convexity, Birkhäuser Verlag, Berlin, 1994 | MR | Zbl

[13] G. M. Khenkin, “Integral Representation Method in Complex Analysis”, Results of Science and Technics. Modern problems of Mathematics. Fundamental Directions, 7, VINITI, Moscow, 1985, 23–124 | MR

[14] A. M. Kytmanov, The Bochner–Martinelli Integral and its Applications, Birkhäuser Verlag, Basael–Boston–Berlin, 1995 | MR | Zbl

[15] Yu. V. Egorov, M. A. Shubin, “Linear Partial Differential equations. Foundations of Classical Theory”, Results of Science and Technics. Modern problems of Mathtmatics. Fundamental Directions, 30, VINITI, Moscow, 1988 | MR

[16] R. A. Adams, Sobolev Spaces, Academic Press. Inc., Boston et al., 1978 | Zbl

[17] A. A. Shlapunov, N. N. Tarkhanov, “Duality by reproducing kernels”, Internat. J. Math. Math. Sci., 2003:6 (2003), 327–395 | DOI | MR | Zbl

[18] E. M. Chirka, “Analytic representation of $CR$-functions”, Mat. Sb., 98(140):4 (1975), 591–623 | MR | Zbl