In Evans book (and many others) there are a classic result about interior regularity in Sobolev spaces for solutions to uniformly elliptic problem (Theorem 1, p. 309). That is, let $\Omega\subset\mathbb{R}^n$ be an open bounded set, $L$ a divergence form operator, that is, for the sake of simplicity $$Lu=-\rm{div}(A(x)\nabla u),$$ where $A=(a_{ij})$ is a symmetric matrix, uniformly elliptic ($\Lambda^{-1}|\xi|^2\le A(x)\xi\cdot\xi\le \Lambda|\xi|^2$ for a.e. $x\in\Omega$ and a constant $\Lambda>0$), with enough regularity of the coefficients $(a_{ij})$. If $u\in H^1(\Omega)$ is a weak solution to $Lu=f$, whit $f\in L^2(\Omega)$ then $u\in H^2(\omega)$ for all $\omega\subset\subset\Omega$ compactly contained, whit the following estimate $$\|u\|_{H^2(\omega)}\le C(\|f\|_{L^2(\Omega)}+\|u\|_{L^2(\Omega)}).$$

I want a reference for an analogous result for a parabolic PDE. In Evans book there is the following theorem (Thm. 5, p. 360) which guarantees higher regularity of solutions up to the boundary, assuming certain hypotheses on the initial data: Let $u\in L^2(0,1;H_0^1(\Omega))$, whit $\partial_t u\in L^2(0,1;H^{-1}(\Omega))$ a weak solution to \begin{cases}u_t+Lu=f &\rm{in }\hspace{0.3cm}\Omega\times(0,1)\\ u=0 & \rm{on }\hspace{0.3cm}\partial \Omega\times(0,1)\\u=g & \rm{on }\hspace{0.3cm}\Omega\times\{0\},\end{cases} where $f\in L^2(0,1;L^2(\Omega))$ and $g\in L^2(\Omega)$. Then we have $u\in L^2(0,1;H^2(\Omega))\cap L^\infty(0,1;H^1_0(\Omega))$, whit $\partial_t u\in L^2(0,1;L^2(\Omega))$ and an estimate of $u$ and $\partial_t u$ in these spaces depending only on the initial data.

I want interior regularity without fixing a boundary condition and initial condition in the spirit of the elliptic theorem, that is considering a compact subset $\omega\subset\subset\Omega$ being able to get estimates in better Sobolev spaces for the solution.

I tried to find this in many other books, but nowhere is this result stated (maybe it's not true, but I don't think so). Any suggestions or references are welcome.


1 Answer 1


You can find such results in Chapter 2, Section 4 of N. Krylov "Lectures on Elliptic and Parabolic equations in Sobolev Spaces" and also in Chapter 5, Section 2 in the $L^p$ setting. Krylov assumes non-divergence operators but this does not matter since your coefficients are regular. However, he starts from strong solutions rather than variational ones and this perhaps requires some adjustment.


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