Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

Question

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$.

Quiz

Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in H^3(\Omega)$.

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Puzzles: trying to break the question into parts...

• It looks like the $H^3$ means the Sobolev space but I am not sure whether this Sobolev space is a Hilbert space: 3 parameters $x,y,z$ while $p=2$ so not a Hilbert space?

• what does the minimality for $\alpha_3$ mean?

• Here: $|...|$ is not an absolute value but the first norm so $|\alpha|\leq s$ does not mean $-s\leq \alpha \leq s$ but the sum of the terms in $\alpha$ is smaller than equal to $s$, correct?

• Here: what does "$\partial_x^\alpha f= \partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}f$ are taken in a weak sense" mean?

Please, help me to understand this quiz. I cannot yet see where the quiz is aiming at, my instructor recommend to use Mathematica but I cannot yet understand this recommendation -- first, please, clarify the Mathematical aspects of the question.

Answer 1

$u$ is explicitly a function of three variables. You should interpret $\Omega\subseteq \mathbb{R}^3 = \{ (x,y,z) \}$.

As usual, $H^3(\Omega) := W^{3,2}(\Omega)$ with the Sobolev norm

$$ \| f\|_{H^3} = \int_{\Omega} \sum_{|\beta| \leq 3} |\partial^\beta f|^2 ~dxdydz $$

where for $\beta$ we use the multi-index notation.

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Away from $x = 0$, $u$ is $C^\infty$, so for any bounded domain $\Omega$ not touching the plane $\{x = 0\}$, the function $u \in H^3(\Omega)$.

near $x = 0$, however, you can explicitly compute the derivatives $\partial_x u$, $\partial^2_x u$ and $\partial^3_x u$ and see where things start breaking as you take smaller and smaller real number $\alpha$.