We show that any harmonic (with respect to the Bergman metric) vector field tangent to the Levi distribution of the foliation by level sets of the defining function $\varphi (z)=- K(z,z)^{-1/(n+1)}$ of a strictly pseudoconvex bounded domain $\Omega \subset \mathbb{C}^n$ which is smooth up to the boundary must vanish on $\partial \Omega$. If $n \neq 5$ and $u T$ is a harmonic vector field with $u \in C^2(\overline{\Omega})$ then $u|_{\partial \Omega} = 0$.

### On the Dirichlet problem for the harmonic vector fields equation

#### Abstract

We show that any harmonic (with respect to the Bergman metric) vector field tangent to the Levi distribution of the foliation by level sets of the defining function $\varphi (z)=- K(z,z)^{-1/(n+1)}$ of a strictly pseudoconvex bounded domain $\Omega \subset \mathbb{C}^n$ which is smooth up to the boundary must vanish on $\partial \Omega$. If $n \neq 5$ and $u T$ is a harmonic vector field with $u \in C^2(\overline{\Omega})$ then $u|_{\partial \Omega} = 0$.
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2007
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11563/6239