On the Cauchy problem for the multi-dimensional Cauchy-Riemann operator in the Lebesgue space~$L^2$ in a~domain

D. P. Fedchenko ; A. A. Shlapunov

Sbornik. Mathematics, Tome 199 (2008) no. 11, pp. 1715-1733

Résumé

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.