Consider
![\[\left(a^{11}+a^{22}\right)u_t - \sum_{i,j=1}^2 a^{ij}(t,x_1,x_2) D_{ij}u = f,\]](https://mathematicians.korea.ac.kr/dkim/wp-content/ql-cache/quicklatex.com-cf566bc2d4ccbdeded391534b113dadd_l3.png "Rendered by QuickLaTeX.com")
where 
are measurable in 
and 
is measurable in 
, but constant in 
.
Set
![\[\left[u_{ij}\right] = \left[ \begin{tabular}{cc} $D_{11} u - u_t$ & $D_{12}u$\\ $D_{21}u$ & $D_{22}u - u_t$ \end{tabular} \right].\]](https://mathematicians.korea.ac.kr/dkim/wp-content/ql-cache/quicklatex.com-8670d1f90be43e7120771e9a7cb31bdf_l3.png "Rendered by QuickLaTeX.com")
We use the inequality
![\[\frac{1}{2\mu^2} \left( \sum a^{ij} u_{ij} \right)^2 \ge \frac{\mu^2}{2\nu^2} \sum u_{ij}^2 + \det \left[u_{ij}\right].\]](https://mathematicians.korea.ac.kr/dkim/wp-content/ql-cache/quicklatex.com-89a10474a1761b9d946dd9f139740f82_l3.png "Rendered by QuickLaTeX.com")
Parabolic equations with measurable coefficients in $\mathbb{R} \times \mathbb{R}^2$
Consider
where are measurable in and is measurable in , but constant in .
Set
We use the inequality