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A simple example of a harmonic field is a vector field $u\in C^1(\Omegabar;\re^3)$ on a domain $\Omega\subset\re^3$ which satisfies
$$
\operatorname{div}u=0\quad\text{ and }\quad\operatorname{curl}u=0\quad\text{ in }\Omega.
$$
If, moreover, $\nu$ is the outward unit normal on the boundary $\partial\Omega$ and $u_N=\langle\nu,u\rangle \nu=0$ on $\partial\Omega,$ we say that $u$ has vanishing normal component $u_N$. I will talk about conditions on $\Omega$ which imply that harmonic fields with vanishing normal or tangential part have to be identically $0.$ More interesting is the case, when $\Omega$ is replaced by a manifold with boundary. For simplicity I will mainly focus on vector fields given on a $2$ dimensional surface embedded in $\re^3,$ but shall also mention the more general case of differential forms on Riemannian manifolds. If time permits, I will give an application in the calculus of variations.
I will make my talk elementary and moreover comprehensible also to those who are not familiar with the language of general Riemannian manifold theory. |