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While studying a nonlinear PDE arising from quantum mechanics, I met a statement that I cannot prove easily. Let us write $E=W^{1,2}(\mathbb{R}^3)$ for the usual Sobolev space, and define the functional $$ \mathcal{D}(u)= \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{|u(x)|^2 |u(y)|^2}{|x-y|} \, dx \, dy \quad \text{for $u \in E$}. $$ It is claimed without proof that $\mathcal{D} \in C^2(E)$. I think I can prove the continuity of the second derivative at zero, but I can't switch to the continuity at different points. I would be grateful for any hint.

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    What did you find as second derivative?2012-09-23
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    The formula for $D^2 \mathcal{D}(u)$ is trivially deduced by formal rules, since $\mathcal{D}$ contains only powers of $u$. My problem is to show that the formula is rigorous, and that it is a continuous function of $u$. This functional is used in Buffoni's paper (http://library.epfl.ch/en/theses/?nr=1035), where the smoothness is stated but not proved.2012-09-23
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    First, how do you see it's well-defined?2012-09-23
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    Buffoni shows that $\int \frac{|u(y)|^2}{|x-y|}dy$ is bounded (in $x$). But it can also be seen as a consequence of some Young inequality for convolutions.2012-09-23
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    In this case, we can rigorously compute the first derivative, expanding $(u(x)+h(x))^2$ and the same for $y$. Then I guess from that we can prove the formula for second derivatives.2012-09-23
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    I am worried by the continuity of the second derivative. For local terms, I often use the theory of superposition operators, but I do not know if I can adapt this strategy, since I cannot find a precise growth condition for the convolution term.2012-09-23

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