Sobolev space $H^2$ norm in terms of gradient

Question

Is it the case that $$|v|_{H^2(\Omega)}^2 = |v|_{H^1(\Omega)}^2 + \int_\Omega\sum_{i=1}^n |\nabla v_{x_i}|^2?$$ I think yes but I have never seen anybody write it like this. I guess generalisations are possible for $H^k$ functions. Is there any nicer way to write the RHS, so I want to make no reference to partial differentiation but gradients are OK..

Answer 1

The two norms are definitely equivalent, and might be equal, depending on your exact definition of $|v|_{H^2(\Omega)}$. To show the equivalence, you might need to use the Cauchy-Schwarz inequality and the fact that $\frac{\partial}{\partial x_i} \left( \frac{\partial v}{\partial x_j} \right) = \frac{\partial^2 v}{\partial x_i \partial x_j}$ which is immediate from the definition (as typically a priori the left hand and the right hand have different definitions).