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In [1] the authors define the function spaces \begin{align*} V(\mathbb{R}^N) = \lbrace v \in \mathbb{R}^N \to \mathbb{C}: ~ &\nabla v \in L^2(\mathbb{R}^N), \\ &\Re(v) \in L^2(\mathbb{R}^N), \\ & \Im(v) \in L^4(\mathbb{R}^N), \\ &\nabla \Re(v) \in L^{4/3}(\mathbb{R}^N) \rbrace \end{align*} and \begin{equation} W(\mathbb{R}^N) = \lbrace 1 \rbrace + V(\mathbb{R}^N). \end{equation}

Later in the same paper they claim that the functional \begin{equation} v \mapsto \int_{\mathbb{R}^N} (1-\vert 1+v \vert^2)^2 \end{equation} is continuous in $V(\mathbb{R}^N)$ and thefore the functional \begin{equation} w \mapsto \frac{1}{2} \int_{\mathbb{R}^N} \vert \nabla w\vert^2 +\frac{1}{4}\int_{\mathbb{R}^N} (1-\vert w \vert^2)^2 \end{equation} is continuous in $W(\mathbb{R}^N)$.

I would like to check this, but I don't even know what the norm on these spaces is supposed to be. So my question is:

What norm do the authors implicitly equip these spaces with to infer the continuity of the above functionals and how do you start checking the continuity?

One idea would be \begin{equation} \Vert v \Vert_V = \Vert \Re(v) \Vert_{L^2} + \Vert \nabla v \Vert_{L^2} + \Vert \Im(v) \Vert_{L^4} + \Vert \nabla \Re(v) \Vert_{L^{4/3}} \end{equation} which indeed is a norm. But what about the norm on $W$?


[1] Béthuel, F., P. Gravejat und J. C. Saut: Travelling waves for the Gross- Pitaevskii equation. II. Comm. Math. Phys., 285(2):567–651, 2009.

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If you look at the proof of Lemma 3.1 of the paper by Béthuel et al., you can easily guess that they use the sum of the natural norms, as you guess.

Finally, the space $W(\mathbb{R}^n)$ is not a vector space, but rather an affine space. Hence you may just notice work with those elements of the form $1+v$ with $v \in V(\mathbb{R}^N)$, and you get an induced topological structure by declaring that $1+v_1$ is "close" to $1+v_2$ if and only if $v_1$ is "close" to $v_2$.

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  • $\begingroup$ Thank you very much! But how does one prove the continuity? For simplicity consider the functional $F(v) = \int \vert Dv \vert^2$. Now if $\Vert v-w \Vert_V < \delta$ in particular we have that $\Vert D v - Dw \Vert_{L^2} < \delta$. But $\vert F(v)-F(w) \vert = \vert \int \vert Dv \vert^2 - \vert Dw \vert^2 \vert$. How does one proceed? Any hint? $\endgroup$ – mjb Jul 25 '13 at 7:36

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