Consider

    \[\left(a^{11}+a^{22}\right)u_t - \sum_{i,j=1}^2 a^{ij}(t,x_1,x_2) D_{ij}u = f,\]

where a^{ij}(t,x_1,x_2) are measurable in (t,x_1,x_2) and a^{11}+a^{22} is measurable in t, but constant in x \in \mathbb{R}^d.

Set

    \[\left[u_{ij}\right] = \left[ \begin{tabular}{cc} $D_{11} u - u_t$ & $D_{12}u$\\ $D_{21}u$ & $D_{22}u - u_t$ \end{tabular} \right].\]

We use the inequality

    \[\frac{1}{2\mu^2} \left( \sum a^{ij} u_{ij} \right)^2 \ge \frac{\mu^2}{2\nu^2} \sum u_{ij}^2 + \det \left[u_{ij}\right].\]

Parabolic equations with measurable coefficients in $\mathbb{R} \times \mathbb{R}^2$