Weak form of the Navier-Cauchy equation

I am trying to obtain the weak form of the Navier-Cauchy equation, which is

$$- rho omega ^2 textbf{U} – mu nabla ^2 textbf{U} – (mu + lambda) nabla (nabla cdot textbf{U}) = textbf{F}$$

and can be written in the component form

$$-(2 mu +lambda) frac{partial ^2 U_1}{partial x_1 ^2} – mu frac{partial ^2 U_1}{partial x_2 ^2} – (mu + lambda) frac{partial ^2 U_2}{partial x_1 partial x_2} – rho omega ^2 U_1 = F_1$$

$$-(2 mu +lambda) frac{partial ^2 U_2}{partial x_2 ^2} – mu frac{partial ^2 U_2}{partial x_1 ^2} – (mu + lambda) frac{partial ^2 U_1}{partial x_1 partial x_2} – rho omega ^2 U_2 = F_2$$

The general procedure is to multiply the PDE by a test function $textbf{v}$ in the space $textbf{V}$, or $v$ in the space $V$, and integrate it over the domain $Omega$. I will proceed with the component form, for I believe it is easier for me to understand. Setting $textbf{F} = 0$ and rearranging the terms

$$-(2 mu +lambda) int_Omega v left[ frac{partial ^2 U_1}{partial x_1 ^2} + frac{partial ^2 U_2}{partial x_2 ^2} right]dxdy – mu int_Omega v left[ frac{partial ^2 U_1}{partial x_2 ^2} + frac{partial ^2 U_2}{partial x_1 ^2} right]dxdy -(mu + lambda)int_Omega v left[ frac{partial ^2 U_2}{partial x_1 partial x_2} + frac{partial ^2 U_1}{partial x_1 partial x_2} right]dxdy – rho omega ^2 int_Omega v left[ U_1+U_2 right]dxdy = 0$$

From Green’s theorem I know that
$$ int_{Omega} left(v frac{partial ^2 u}{partial x ^2} right)dxdy = int_Gamma left(v frac{partial u}{partial x} hat{n}_x right)ds – int_{Omega} left( frac{partial v}{partial x} frac{partial u}{partial x} right)dxdy$$

Which is sufficient to deal with the first and second integrals. However, I do not know how to proceed with the cross derivatives $partial ^2 / partial x_1 partial x_2$ of the third integral. Can someone help me with this?