Abstract:
In two dimensions, harmonic functions share their of properties with holomorphic functions. Some of these properties still hold in higher dimensions, like e.g. the mean value property or analyticity.

One of the properties is that there exists no harmonic function on the two dimensional upper half plane, such that the function and it's normal derivative both vanish on a boundary set of harmonic measure.
We will construct such a function in higher dimensions.
This proof dates back to Bourgain and T. Wolff and can be modified to show similar results under broader assumptions.