# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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### Doubt obtaining the inequality (3) via (1)

I am reading a proof from the 2009-paper Travelling waves for the Gross-Pitevskii equation II (Béthuel, Gravejat, Saut) and I am really stuck in one step. Please, help me with this: In the paper, ...
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### Alternative form for Liapunov inequality

Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then $$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$ My teacher ...
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### Functional is weak lower semicontinuous but not weak continuous

I want to show that the functional $L(u)=\int_0^1 \sqrt{1+(u'(x))^2} dx$ is lower semicontinuous in terms of weak convergence in $W^{1,p}(0,1), p\in(1,\infty)$ but not continuous. Our definition of ...
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