Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that

$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$

Further assume a solvability condition

$$\int_\Omega f ~\mathrm{d}\mathbf{x} = 0.$$

It is known, that this problem is solvable up to a constant. So speaking in terms of operators this means that the operator $A\colon H^1(\Omega) \to \widetilde{H}^{-1}(\Omega)$ defined by $$(A u, v) := \int_\Omega \nabla u \nabla v ~ \mathrm{d}\mathbf{x}$$ is not invertible. However one can always build up the moore-penrose pseudoinverse $A^\dagger$. So the minimum-norm-solution is given by $$u = A^\dagger f$$.

My question is, whether $A^\dagger$ is also bounded and maybe coercive (or semi-elliptic) ?

Thanks!