integral equality assumptions I would like to prove the following equality
$$\int_{\mathbb{R}^2}\frac{\partial^2 u(x,y)}{\partial x^2}\frac{\partial^2 v(x,y)}{\partial x\partial y}dxdy=\int_{\mathbb{R}^2}\frac{\partial^2 u(x,y)}{\partial x\partial y}\frac{\partial^2 v(x,y)}{\partial x^2}dxdy$$
What is known that all above derivatives exist, u has compact support and the derivatives of v are bounded.
What the least needs to be assumed to be able to prove the equality? I would like to avert the higher derivatives of at least one of the functions (preferably v).
 A: Formally,
$$\intop_{\mathbb{R}^{2}}\frac{\partial^{2}u\left(x,y\right)}{\partial x^{2}}\frac{\partial^{2}v\left(x,y\right)}{\partial x\partial y}dxdy=\intop_{x\in\mathbb{R}}\left[\intop_{y\in\mathbb{R}}\frac{\partial^{2}u\left(x,y\right)}{\partial x^{2}}\frac{\partial^{2}v\left(x,y\right)}{\partial x\partial y}dy\right]dx
 $$
 $$=\intop_{x\in\mathbb{R}}\left[\frac{\partial^{2}u\left(x,y\right)}{\partial x^{2}}\frac{\partial v\left(x,y\right)}{\partial x}\right]_{y\in\mathbb{R}}dx-\intop_{x\in\mathbb{R}}\left[\intop_{y\in\mathbb{R}}\frac{\partial^{3}u\left(x,y\right)}{\partial y\partial x^{2}}\frac{\partial v\left(x,y\right)}{\partial x}dy\right]dx
 $$
 $$=\intop_{x\in\mathbb{R}}\left[\frac{\partial^{2}u\left(x,y\right)}{\partial x^{2}}\frac{\partial v\left(x,y\right)}{\partial x}\right]_{y\in\mathbb{R}}dx-\intop_{y\in\mathbb{R}}\left[\intop_{x\in\mathbb{R}}\frac{\partial^{3}u\left(x,y\right)}{\partial x\partial y\partial x}\frac{\partial v\left(x,y\right)}{\partial x}dx\right]dy
 $$
 $$=\intop_{x\in\mathbb{R}}\left[\frac{\partial^{2}u\left(x,y\right)}{\partial x^{2}}\frac{\partial v\left(x,y\right)}{\partial x}\right]_{y\in\mathbb{R}}dx-\intop_{y\in\mathbb{R}}\left[\frac{\partial^{2}u\left(x,y\right)}{\partial y\partial x}\frac{\partial v\left(x,y\right)}{\partial x}\right]_{x\in\mathbb{R}}dy+\intop_{y\in\mathbb{R}}\left[\intop_{x\in\mathbb{R}}\frac{\partial^{2}u\left(x,y\right)}{\partial y\partial x}\frac{\partial^{2}v\left(x,y\right)}{\partial x^{2}}dx\right]dy
 $$
 $$=\intop_{x\in\mathbb{R}}\left[\frac{\partial^{2}u\left(x,y\right)}{\partial x^{2}}\frac{\partial v\left(x,y\right)}{\partial x}\right]_{y\in\mathbb{R}}dx-\intop_{y\in\mathbb{R}}\left[\frac{\partial^{2}u\left(x,y\right)}{\partial y\partial x}\frac{\partial v\left(x,y\right)}{\partial x}\right]_{x\in\mathbb{R}}dy+\intop_{\mathbb{R}^{2}}\frac{\partial^{2}u\left(x,y\right)}{\partial x\partial y}\frac{\partial^{2}v\left(x,y\right)}{\partial x^{2}}dxdy.
 $$
I used Fubini theorem (to permute integrals), integrations by part and Schwarz theorem (to permute partial derivatives). Now you must check that it was allowed.
