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For a given $u \in L^2(\mathbb{R}^N)$, there is a unique $S(u) \in H^1(\mathbb{R}^N)$ such that $$ \int_{\mathbb{R}^N} \nabla S(u) \nabla \varphi + S(u) \varphi = \int_{\mathbb{R}^N} u \varphi, \quad \forall \varphi \in H^1_{\mathbb{R}^N}. $$ I want to prove that, for $u \in L^2(\mathbb{R}^N)$, the following inequality holds $$ (1) \,\,\,\,\, \int_{\mathbb{R}^N} u^{+} S(u^+) \leq \int_{\mathbb{R}^N} u S(u). $$ What I know and can be used: $\int_{\mathbb{R}^N} u S(u) \geq 0$ for any $u \in L^2(\mathbb{R}^N)$ and $\int_{\mathbb{R}^N} u S(v) = \int_{\mathbb{R}^N} v S(u)$ for any $u,v \in L^2(\mathbb{R}^N)$.

I know how to prove $(1)$ assuming two situations, which I don't know it if true or not. The first one is assuming $u(x)S(u^+)(x) \geq 0$ for $x \in \{x \in \mathbb{R}^N : u(x) < 0\}$. The second one is assuming $\int_{\mathbb{R}^N} u^-S(u^+) = 0$. Some of this two conditions are true in fact?

ThiagoGM
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1 Answers1

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I think this inequality cannot be true. Here is a counterexample for $N=1$:

Define $u_n(x) := \sin(2n\pi x) \chi_{[0,1]}$. Then $u_n \rightharpoonup 0$ in $L^2(0,1)$ and in $L^2(\mathbb R)$. The sequence $(S(u_n))$ is bounded in $H^1(\mathbb R)$ and $H^1(0,1)$, and there is a subsequence converging strongly in $L^2(0,1)$. This implies $\int_{\mathbb R} u_n S(u_n) =\int_0^1 u_nS(u_n)\to 0$.

Now $u_n^+ \rightharpoonup v:=\chi_{[0,1]}c$, where $c = \int_0^1 (\sin(2\pi x))^+ dx>0$. This implies $S(v) \ne0$, and $$ \int_{\mathbb R} S(v)v = \|S(v)\|_{H^1(\mathbb R)}^2 >0. $$ Similarly to the arguments in the first paragraph, we get $\int_{\mathbb R} u_n^+ S(u_n^+) \to \int_{\mathbb R} v S(v) >0$.

So the right-hand side of the desired inequality tends to zero, while the left-hand side tends to some positive number. And the inequality cannot be true.

(The same arguments show that the inequality cannot be true if we replace $\mathbb R^N$ by any domain $\Omega \subset \mathbb R^N$.)

daw
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  • Do you know how to prove that $\int_{\mathbb{R}^N} u^- S(u^+) = 0 ?$ This would help me because $\int_{\mathbb{R}N} u S(u) = \int_{\mathbb{R}N} u^+ S(u^+) + \int_{\mathbb{R}N} u^- S(u^-) - 2 \int_{\mathbb{R}N} u^- S(u^+)$. – ThiagoGM Apr 24 '24 at 14:55
  • If your answer is true, then is it impossible to have $\int_{\mathbb{R}^N} u^- S(u^+) = 0$, right? – ThiagoGM Apr 24 '24 at 14:57
  • Yes, this integral cannot be zero then. – daw Apr 24 '24 at 15:30
  • Thank you for the answer. I didn't understand why $\int_0^1 u_nS(u_n)\to 0$. Could you, please, clarify it ? I have no ideia, because the norm of $u_n$ in $L^2$ is constant for all $n$. – ThiagoGM Apr 24 '24 at 20:18
  • For weak convergence of $u_n$, see https://math.stackexchange.com/questions/3536412/how-to-show-weak-convergence-of-mathrmsign-sinnx-as-n-to-infty?noredirect=1&lq=1 – daw Apr 25 '24 at 06:24