# Relation of $\lVert \dot u \rVert_{W^1(\Omega)}$ to $\lVert u \rVert_{W^1(\Omega)}$

I am working with the space $W(\Omega) = H^1(\Omega)/\mathbb{R}$. I have the following definition and proposition:

1. 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 $u$.

2. 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!

• Apologies - I misunderstood your previous question. (You should leave a comment if this happens, rather than asking a new question!) What is your definition of $|| u ||_{W^1(\Omega)}$? – Kenny Wong May 19 '17 at 13:29
• That is what I wanted to know. I am using the book Introduction to Homogenization by Cioranescu, Damlamian, and Griso. They did not define there $\lVert u \rVert_{W(\Omega)}$ but they used it in the a priori estimate. The norm defined in the book is $\lVert \dot{u} \rVert_{W(\Omega)}$. Sorry there was a typo. I am considering the space $W(\Omega)$ not $W^1(\Omega)$. I'll keep in mind what you said about leaving comments. Thanks! – stonehenge May 19 '17 at 13:41
• My best guess is that $|| u ||_{W(\Omega)}$ refers to $|| u ||_{H^1(\Omega)} = \left( || u ||_{L^2 (\Omega)}^2 + || \nabla u ||_{L^2(\Omega)}^2 \right)^{\frac 1 2}$. This to me seems like the only thing that would make sense in the context of your previous question on math.SE, assuming that the motivation for your previous question is to prove existence of weak solutions to elliptic PDEs. I don't have your book I'm afraid, so what I'm saying is based on what I've learned from Evan's book. – Kenny Wong May 19 '17 at 13:47
• Yes my goal is indeed to prove the existence of a weak solution of a PDE with a nonhomogeneous Neumann boundary condition. You think I can use the same norm for $W(\Omega)$? I will define the norm $\lVert u \rVert_{W(\Omega)}$ = $\lVert u \rVert_{H^1(\Omega)}$? Also what is the title of Evan's book? Many thanks! – stonehenge May 19 '17 at 13:52
• Well $|| . ||_{H^1(\Omega)}$ cannot be used as a norm on elements of $W(\Omega)$ because the value of $|| . ||_{H^1(\Omega)}$ depends on which representative you pick for the equivalence class. The answer I gave was really intended for Dirichlet boundary conditions, where $u$ is assumed to be in $H^1_0(\Omega)$ (the zero in the subscript symbolising that $u$ is in the closure of the space of functions with compact support). – Kenny Wong May 19 '17 at 14:03

## 1 Answer

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.