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Let $h\in L^2(0,1)$, such that |\int h \mathrm{d}x|<1. On the space $H^1(0,1)$, consider $J(v)=\frac{1}{2}\int_0^1 v'^2\,\mathrm{d}x+\int_0^1\sqrt{1+v^2}\,\mathrm{d}x-\int_0^1hv \,\mathrm{d}x.$
How to show that $J(u)\rightarrow +\infty$ when $\|v\|_{H^1}\rightarrow +\infty$?

I'm stuck. There is a hint: $\forall v\in H^1(0,1), \|v-\int_0^1 v\,\mathrm{d}x\|_{L^\infty}\leqslant\int_0^1|v'|\,\mathrm{d}x,$ but I don't see how it helps. Could anyone please help?

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    Poincaré inequality is valid for $v\in H^1_0$. How to solve it if $v\in H^1$? – 2012-04-26

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