Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if $$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 $$ Where $Q$ is $[0,T]^n$ and $C>0$ is a constant. Is there some Gagliardo-Niremberg inequality for $H^1_T$-functions? Please let me know. Thanks in advance!

  • $\begingroup$ Why do you have a $1$ on the right-hand side? What happens if $u$ is a constant (different from $1$)? $\endgroup$ – gerw May 21 at 5:53
  • $\begingroup$ What happens if you replace $u$ by $c \, u$ (with a constant $c$) and $c \to \infty$? $\endgroup$ – gerw May 21 at 5:53
  • $\begingroup$ I need to add a constante. True. I made an edit $\endgroup$ – Senna May 21 at 8:37
  • $\begingroup$ So there exists a functional inequality like that? $\endgroup$ – Senna May 21 at 8:38
  • $\begingroup$ The constant does not help with the issues I raised above... $\endgroup$ – gerw May 21 at 11:07

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