I am working with the space $W(\Omega) = H^1(\Omega)/\mathbb{R}$. I have the following definition and proposition:
Suppose that $\Omega$ is connected. The quotient space $$ W(\Omega) = H^1(\Omega)/ \mathbb{R}$$ is defined as the space of classes of equivalence with respect to the relation $$ u \simeq v \Longleftrightarrow u-v \text{ is a constant}, \quad \forall u,v \in H^1(\Omega).$$ We denote by $\dot u$ the class of equivalence represented by $u$.
Suppose that $\Omega$ is connected. The following quantity: $$\lVert \dot{u} \rVert_{W(\Omega)} = \lVert \nabla u \rVert_{L^2(\Omega)}, \quad \forall u \in \dot{u}, \, \dot{u} \in W(\Omega),$$ defines a norm on $W(\Omega)$ for which $W(\Omega)$ is a Banach space. Moreover, $W(\Omega)$ is a Hilbert space for the scalar product $$ (v,w)_{W(\Omega)} = \sum_{i=1}^N \Bigg( \dfrac{\partial v}{\partial x_i} \dfrac{\partial w}{\partial x_i} \Bigg)_{L^2(\Omega)}, \quad \forall v,w \in W(\Omega).$$
My question is: how can I relate $\lVert \dot u \rVert_{W(\Omega)}$ to $ \lVert u \rVert_{W(\Omega)}$?
Thank you!
If $\Omega$ is bounded, connected and sufficiently regular (say $\partial \Omega$ Lipschitz continuous), then you have Poincare-Wirtinger Inequality $$\int_\Omega |u(x)-u_\Omega|^2dx\le c\int_\Omega |\nabla u(x)|^2dx,$$ where $u_\Omega$ is the average of $u$ over $\Omega$. This gives you that $$\inf_{d\in\mathbb{R}}\int_\Omega |u(x)-d|^2dx\le c\int_\Omega |\nabla u(x)|^2dx.$$ Hence in this case the two norms are equivalent. However, if $\Omega$ is not regular (the typical example is called the room and corridors) then in general you do not have Poincare-Wirtinger and so the two norms are not equivalent.