Sobolev embeddings: counter example

Question

let $\Omega =\{(x,y)\in \mathbb{R}^2 : 0 < x < 1, 0 < y < x^2\}$ be a bounded and open domain and it is not $C^1$. And consider the function $u(x,y)=x^a.$

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Use the function $u$ to prove that $H^1(\Omega)$ dose not inject in $L^p(\Omega)$ for $p > 6$.

Answer 1

Notice that $$ \begin{align} \|u\|_{H^1}^2 = &\ \int_0^1\int_0^{x^2}x^{2a}\,dy\,dx + \int_0^1\int_0^{x^2}a^2x^{2a-2}\,dy\,dx\\ = &\ \int_0^1x^{2a + 2} + a^2x^{2a}\,dx, \end{align}$$ so that $u \in H^1(\Omega)$ if and only if $a > -\frac{1}{2}.$ On the other hand, $$\|u\|_{L^p}^p = \int_0^1\int_0^{x^2}x^{pa}\,dy\,dx = \int_0^1x^{pa + 2}\,dx$$ and hence $u \in L^p$ if and only if $pa + 2 > -1$, i.e. $a > -\frac{3}{p}$.

To conclude it is enough to notice that for any $p > 6$ we can find $a$ such that $$-\frac{1}{2} < a < -\frac{3}{p}.$$