Let $\Omega = B(0,1) \subset \mathbb R^n$. Given $f \in C(\partial \Omega)$ (though I guess what I want to do may make more sense in some other space of functions), we can solve the Dirichlet problem for $u_f$ with boundary condition $f$: $$ \Delta u_f(x) = 0, x\in\Omega, $$ $$ u_f(s) = f(s), s \in \partial\Omega $$ which has a unique solution.
Then we can consider the Dirichlet energy of $u$ $$ I(f) := I[u_f] = \int_{\Omega} \vert \nabla u_f \vert^2. $$ Using this procedure we have a map from the space of boundary conditions to $\mathbb R$.
This map is clearly not linear but it satisfies $$ I(f+g) = \int_{\Omega} \vert \nabla u_{f+g} \vert^2 = \int_{\Omega} \vert \nabla u_f + \nabla u_g \vert^2 \leq 2 I(f) + 2I(g). $$ The natural way to study this map is (probably) using the representation of solutions using the Poisson kernel but I haven't had much success. Also, by using the divergence theorem, we have that $2\int_\Omega \vert \nabla u \vert^2 = \int_\Omega \Delta(u^2) = \int_{\partial\Omega} \nabla(u^2) \cdot \nu = 2\int_{\partial\Omega} f \nabla u \cdot \nu$.
I have several questions about this map
- Is this map continuous? I guess it depends on the space of boundary conditions I choose so which one is the natural candidate?
- Can I control from below and above this quantity using some norm? Clearly, $L^p$ norms are not enough since constant functions $f$ have $I(f) = 0$.
- Does anyone have some reference on something similar to this?