The $\bar\partial$ Neumann problem for smooth functions and distributions

A. M. Kytmanov

Sbornik. Mathematics, Tome 70 (1991) no. 1, pp. 79-92

Résumé

We consider the following $\bar\partial$-Neumann problem for functions: given a function $\varphi$ on the boundary of a domain $D\subset\mathbf C^n$ with boundary of class $C^\infty$, find a harmonic function $F$ in $D$ such that $\bar\partial_nF=\varphi$ on $\partial D$ (where $\bar\partial_nF$ is the normal part of the differential form $\bar\partial F$). It is shown that with the homogeneous boundary condition $\bar\partial_nF=0$, the only solutions of this problem are holomorphic functions. Solvability of this problem is proved in strictly pseudoconvex domains if the function (or distribution) $\varphi$ is orthogonal to holomorphic functions $f$ for integration over $\partial D$. An integral formula for the solution of the $\bar\partial$-Neumann problem in the ball is given. The proof uses known results on solvability of the $\bar\partial$-Neumann problem for forms of type $(p,q)$ for $q>0$.