Let f be a $C^3$ function, $f: \mathbb{R}^n \to \mathbb{R}$, f is compactly supported

prove that:

$$\int_{\mathbb{R}^n} (\Delta f) ^2 = \sum_{i,j = 1}^n \int_{\mathbb{R}^n} (\frac{\partial^2f}{ \partial x_i \partial x_j})^2$$

I tried to use green formulas but didnt really know how to handle the laplacian squared

I think it might be realated to the co-area formula by using balls

  • $\begingroup$ Are you sure about the statement? For example, with $f(x_1,x_2)=x_1x_2$ the left-hand side is zero and the right-hand side is $+\infty$. $\endgroup$
    – mickep
    Jun 27, 2016 at 12:26
  • $\begingroup$ you are right, I updated the question, f is compactly supported $\endgroup$ Jun 27, 2016 at 12:31

1 Answer 1


A physicsists idea: On the one hand $$\int dV (\Delta f)^2 = \int dV \left(\sum_i \partial_i^2 f\right)^2= \int dV \left(\sum_i \partial_i^2f\right)\left(\sum_j \partial_j^2f\right)\\=\sum_{i,j} \int dV (\partial_i^2f)(\partial_j^2 f)$$ On On the other hand $$\int dV \sum_{i,j} \left(\partial_i \partial_j f \right)^2=\int dV \sum_{i,j} \left(\partial_i \partial_j f \right) \left(\partial_i \partial_j f \right)\\ =\sum_{i,j} \int dV \left(\partial_i \partial_j f \right) \left(\partial_i \partial_j f \right)$$ Let's look at the last expression. We use partial integration to move a $\partial_i$ from the right parenthesis to the left. We will pick up a minus sign. Next, we use partial integration to move a $\partial_j$ from the left parenthesis to the right parenthesis. We will pick up another minus sign and have shown the identity.

EDIT: I guess that the surface terms from the partial integration vanish due to the compact support of $f$. The appearance of the third derivative after the first partial integration should also not be a problem, since you specified that $f$ is a $C^3$ function.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.