# Regarding regularity of solution of the Heat equation, Brezis' Functional Analysis Theorem 10.1

I have started reading Chapter 10 of Brezis' Functional Analysis and run into a problem in the proof of Theorem 10.1. I have a question regarding the part where it's shown that $A$ is maximal monotone.(Pages attached below)

Theorem 10.1 and Theorem 9.25

So far I have figured this much. Here $\Omega$ is of class $C^{\infty}$ with $\Gamma = \partial \Omega$ bounded. Take any $f \in L^2(\Omega)$. We have by a previous theorem that $\exists$ a unique weak solution to $$u - \Delta u =f \text{ in } \Omega$$ $$u = 0 \text{ on } \Gamma = \partial \Omega$$ That is, $u \in H_{0}^1(\Omega)$ and $$\int_{\Omega} \nabla u \cdot \nabla \phi + \int_{\Omega}u \phi = \int_{\Omega}f \phi \ \ \ \forall \phi \in H_{0}^1(\Omega)$$ Then by Theorem 9.25, $u \in H^2(\Omega)$ and $\left \| u \right \|_{H^2} \leq C \left \| f \right \|_{L^2}$.

From here on how can we proceed to show that the weak solution $u$ actually turns out to be a classical solution? This has been shown to be true if $u \in C^2(\bar{\Omega})$ as in the image below: Classical solution from weak solution

But I can't see if that applies here or how. I feel like this might be an easy step, but I seem to be stuck. Any help is greatly appreciated  ## 1 Answer

The weak solution $u$ is not necessarily classical in this case. Since $f\in L^2$ we conclude that $u \in H^2(\Omega)$. For $u\in C^2(\bar{\Omega})$ we require that $u-\Delta u \in C(\Omega)$ but without additional a priori estimates on $f$ there is no way to improve regularity on $u$. As Theorem 9.25 states, Sobolev embeddings will ensure that $H^m(\Omega)\subset C(\Omega)$ provided $m$ is sufficiently large.

From the information you provided, in the proof that $A$ is maximal monotone it seems it is only required that $u\in H^2(\Omega)\cap H^1_0(\Omega)$ and not $u\in C^2(\bar{\Omega})$.