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Let $\Omega$ be a sufficiently smooth domain, $T>0$, and $L$ be the following elliptic operator of the divergence form: $$Lu(t,x)=a^{ij}(t,x)u_{ij}(t,x),$$ such that $a^{ij}\in C([0,T]\times\bar{\Omega})$ and satisfies the uniform elliptic condition.

I was wondering whether there is a reference showing that, there exist constants $C$ and $\lambda_0$, such that for all $\lambda\ge \lambda_0$ and for all
$$ u\in L^2((0,T);H^2(\Omega))\cap H^1((0,T);L^2(\Omega)):= H^{2,1}(\Omega_T) $$ with $u(0,\cdot)=0$ and $u=0$ on $\partial \Omega$, we have the estimate: $$ \|u\|_{H^{2,1}(\Omega_T)}+\lambda\|u\|_{L^2(\Omega_T)}\le C\|\partial_t u-Lu+\lambda u\|_{L^2(\Omega_T)}. $$ where we denote $\Omega_T=(0,T)\times \Omega$.


For elliptic equations, a similar estimate has been shown in Theorem 9.14 of Gilbarg's book for $Lu(x)=a^{ij}(x)u_{ij}(x)$ under the assumption that $a^{ij}\in C(\bar{\Omega})$. For parabolic equation on the whole space $\mathbb{R}^n$, the estimate has been shown in Krylov's book. Hence I guess a similar estimate for parabolic equations on a bounded domain may already been shown somewhere.

I think it suffices to show $$ \lambda\|u\|_{L^2(\Omega_T)}\le C\|\partial_t u-Lu+\lambda u\|_{L^2(\Omega_T)}. $$ Since then by letting $f=\partial_t u-Lu+\lambda u$, we can estimate the $H^{2,1}$-norm by $$ \|u\|_{H^{2,1}(\Omega_T)}\le C\|\partial_t u-Lu\|_{L^2(\Omega_T)}\le C(\|f\|_{L^2(\Omega_T)}+\lambda \|u\|_{L^2(\Omega_T)})\le C\|f\|_{L^2(\Omega_T)}. $$ However, I am not sure how to derive the $L^2$ estimate, since $L$ is of the nondegenerate form.

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