Let $f,g \in \mathbb{H^2_0(\Omega)}$ and $T=(U.\nabla)$ with $U=(\frac{-\partial (f-g)}{\partial y},\frac{\partial (f-g)}{\partial x})$ I want t prove that: $$\int_{\Omega}T(\Delta f) (f-g)dx=-\int_{\Omega}\Delta f T(f-g)dx$$
My idea is that we can say that $T$ is a distribution and we derive in the distribution sense