I am reading Evans' Partial Differential Equations and encountered something I don't understand:
Let $u \in H^1(U)$ be a weak solution to a second order elliptic PDE with nice enough coefficients $Lu = f,$ for $f\in L^2(U).$ Then, it is shown that $u\in H^2_{\mathrm{loc}}(U).$ The text then claims that this fact implies that $Lu = f$ pointwise a.e. in $U$. Why is this true?
Since $u \in H^2_{\mathrm{loc}}(U)$ you can put all the derivatives on it to rewrite the weak formulation as \begin{align} \int_{U} v \, (L u-f)\, \mathrm{d}x=0 \, , \end{align} for all $v \in H^1_0(U)$ (I am assuming you are working with Dirichlet bcs). Then, by the fundamental lemma of the calculus of variations (see https://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations#Multivariable_functions) we have that $Lu-f=0$ a.e. in $U$.