I am considering the following integral $$ \int_{\mathbb{R}^d} ( [\nabla a ]\cdot \nabla b ) \Delta c \ dx, $$ with $a, b$ and $c$ vanishing in the infinity.
By considering integration by parts, I was wondering if the following is true:
$$ \int_{\mathbb{R}^d} ( [\nabla a ]\cdot \nabla b ) \Delta c \ dx = - \int_{\mathbb{R}^d} (\Delta a)b \Delta c \ dx. $$
Do you have any idea if this could be true o if there exist a way to deal with it ?
No, but the following is true and follows from an application of the first of Green's identities, which are essentially some of the most straightforward generalization of integration by parts to higher dimensions: $$\int_{\mathbb{R}^d}(\nabla a\cdot\nabla b)\Delta c~\mathrm{d}x = -\int_{\mathbb{R}^d} b\nabla\cdot(\Delta c \nabla a)~\mathrm{d}x.$$ This can be expanded more via the product rule to get a $\Delta a$ term if you would like. Also, it is generally true then when integration by parts and related theorems are applied, one derivative is transfered from one function onto all functions multiplying it. In particular, the derivative removed from $b$ must be applied to both $a$ and $c$, so there must be some terms containing 3rd derivatives of $c$ after IBP has been applied.