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

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 $\mathbb{R} \times \mathbb{R}^2$