Assume $u: \mathbb{R}^N \to \mathbb{R}$ is a smooth function with suitable integrability assumptions. I'm interested in a formal computation, do not worry about integrability properties or smoothness of $u$.

Let $a$ be a constant.

By integration by parts, how can one prove that the identity $$\int_{\mathbb{R}^N}u^a\nabla u\cdot\nabla\Delta u=C(\int_{\mathbb{R}^N}|D^2 (u^{(a+2)/2})|^2+\int_{\mathbb{R}^N}|\nabla(u^{(a+2)/4})|^4)$$ holds for $C$ some constant that depends on $a$?

For which $a$ does the previous result hold?

  • $\begingroup$ @AlexFrancisco Unsuccessful integration by parts. $\endgroup$ – Dal Jun 10 '18 at 11:01
  • $\begingroup$ What's $D^2$? It doesn't seem to be Hessian since there's a (vector) norm surrounding it. $\endgroup$ – Saad Jun 13 '18 at 11:15
  • $\begingroup$ @AlexFrancisco No, that's to be intended (with a little of abuse of notation) as the $L^2$ matrix norm $\endgroup$ – Dal Jun 13 '18 at 11:19
  • $\begingroup$ What is $\alpha,$ and why should $u^\alpha$ even be well defined? Also, $u$ is any smooth function? No compact support? $\endgroup$ – zhw. Jun 15 '18 at 15:18
  • $\begingroup$ @zhw. You may assume the right integrability assuptions on $u$. Actually all I need is a formal computation. $\endgroup$ – Dal Jun 15 '18 at 18:16

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