# Evans' PDE -- mean-value property for heat equation

To quote from Evans' PDE book pg 53,

Let $u \in C_1^2(U_T)$ solve the heat equation. Then $$u(x, t) = \frac{1}{4 r^n} \int \int_{E(x, t; r)} u(y, s) \frac{|x - y|^2}{(t - s)^2} dy ds.$$

In the proof of this theorem, the book says, "Upon mollifying if necessary, we may assume $u$ is smooth." I'm not sure how this is done (do we just replace $u$ everywhere with $u^\epsilon$? But then does the theorem statement have to be revised to have $u^\epsilon$ instead of $u$?) or why this is necessary (we already are given that $u \in C_1^2 (U_T)$).

You prove the result for $u^{\epsilon}$ for all $\epsilon > 0$. Since $u^{\epsilon} \to u$ uniformly (See Theorem 7 of appendix) and $E(x,t;r) \subset U_{T}$ where $U_{T}$ is bounded, uniform convergence allows you to prove the statement for $u$ given that it is true for $u^{\epsilon}$ for all $\epsilon > 0$.
• It is important to notice that $u^\varepsilon$ also satisfies the Heat equation Commented Oct 12, 2021 at 4:27