It is often stated that the Dirichlet problem on a domain (say a disk in the plane) can be solved by finding the extension of the boundary data with minimal Dirichlet energy.
However, this obviously only makes sense if there is an extension of finite energy to begin with. You can find an example, due to Hadamard, where this is not the case in these notes (page 9).
Are there any good conditions on the boundary function to ensure that this is the case? For example, do the following classes of functions have finite energy extensions: smooth (yes), Lipschitz, Holder...? What about something like Brownian motion?
Alternatively, are there any standard `candidate functions' that tend to give good bounds for the energy?
For example, we could take the homogenous extension $u(rz)=r^{\alpha}f(z)$ and one can show that the dirichlet energy of this extension is bounded by the energy of the boundary data. This is far too weak to answer my question for Brownian boundary data though.