Geodesic
On the Cauchy problem for the multi-dimensional Cauchy-Riemann operator in the Lebesgue space $L^2$ in a domain
Sbornik. Mathematics, Tome 199 (2008) no. 11, pp. 1715-1733 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $D$ be a bounded domain in $\mathbb C^n$ ($n\geqslant1$) with infinitely smooth boundary $\partial D$. We describe necessary and sufficient conditions for the solvability of the Cauchy problem in the Lebesgue space $L^2(D)$ in the domain $D$ for the multi-dimensional Cauchy-Riemann operator $\overline\partial$. As an example we consider the situation where the domain $D$ is the part of a spherical shell $\Omega(r,R)=B(R)\setminus\overline B(r)$, $0, in $\mathbb C^n$, where $B(R)$ is the ball of radius $R$ with centre at the origin, cut off by a smooth hypersurface $\Gamma$ with the same orientation as $\partial D$. In this case, using the Laurent expansion for harmonic functions in the shell $\Omega(R,r)$ we construct the Carleman formula for recovering a function in the Lebesgue space $L^2(D)$ from its values on $\overline\Gamma$ and the values of $\overline\partial u$ in the domain $D$, if these values belong to $L^2(\Gamma)$ and $L^2(D)$, respectively. Bibliography: 16 titles. 
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D. P. Fedchenko; A. A. Shlapunov. On the Cauchy problem for the multi-dimensional Cauchy-Riemann operator in the Lebesgue space $L^2$ in a domain. Sbornik. Mathematics, Tome 199 (2008) no. 11, pp. 1715-1733. http://geodesic.mathdoc.fr/item/SM_2008_199_11_a5/

[1] T. Carleman, Les fonctions quasi analytiques, Gauthier-Villars, Paris, 1926 | Zbl

[2] L. Aizenberg, Carleman's formulas in complex analysis. Theory and applications, Math. Appl., 224, Kluwer Acad. Publ., Dordrecht, 1993 | MR | MR | Zbl | Zbl

[3] L. A. Aǐzenberg , A. M. Kytmanov, “On the possibility of holomorphic extension into a domain of functions defined on a connected piece of its boundary”, Math. USSR-Sb., 72:2 (1992), 467–483 | DOI | MR | Zbl | Zbl

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

[5] N. N. Tarkhanov, The Cauchy problem for solutions of elliptic equations, Math. Top., 7, Academie-Verlag, Berlin, 1995 | MR | Zbl

[6] G. M. Khenkin, “The method of integral representations in complex analysis”, Several complex analysis. I. Introduction to complex analysis, Encyclopaedia Math. Sci., 7, Springer-Verlag, Berlin, 1990, 19–116 | MR | Zbl

[7] Yu. V. Egorov, M. A. Shubin, “Linear partial differential equations. Foundations of the classical theory”, Partial differential equations, I, Encyclopaedia Math. Sci., 30, Springer-Verlag, Berlin, 1992, 1–259 | MR | MR | Zbl

[8] A. M. Kytmanov, The Bochner–Martinelli integral and its applications, Birkhaüser, Basel, 1995 | MR | Zbl | Zbl

[9] N. Kerzman, “Hölder and $L^p$ estimates for solutions of $\overline\partial u=f$ in strongly pseudoconvex domains”, Comm. Pure Appl. Math., 24:3 (1971), 301–379 | DOI | MR | Zbl

[10] L. Hörmander, “$L^2$ estimates and existence theorems for the $\overline\partial$ operator”, Acta Math., 113:1 (1965), 89–152 | DOI | MR | Zbl

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

[12] V. P. Mikhajlov, Partial differential equations, Mir, Moscow, 1978 | MR | MR | Zbl | Zbl

[13] E. M. Čirka, “Analytic representation of $CR$-functions”, Math. USSR-Sb., 27:4 (1975), 526–553 | DOI | MR | Zbl

[14] H. S. Shapiro, “Stefan Bergman's theory of doubly-orthogonal functions. An operator-theoretic approach”, Proc. Roy. Irish Acad. Sect. A, 79:6 (1979), 49–58 | MR | Zbl

[15] S. L. Sobolev, Cubature formulas and modern analysis: An introduction, Gordon and Breach, Montreux, 1992 | MR | MR | Zbl | Zbl

[16] N. N. Tarkhanov, Ryad Lorana dlya reshenii ellipticheskikh sistem, Nauka, Novosibirsk, 1991 | MR | Zbl