Definition 12.1.3 (Linear Function).label We say the function $F(x,\parens{\pd^\al u}_{\abs{\al}\leq k})$ is linear (in $u$) if it is equivalent to
\begin{align*}\sums{}{\abs{\al}\leq k}a_{\al}(x)\pd^{\al} u=b(x)\end{align*}
For linear equations with coefficients $a_{\al}\in C^{\infty}(\R^{n})$ one can also talk about weak solutions. We write $\inner{f}{g}\define \integral{\R^n}{}f g dx$ if $f,g$ are integrable. $u$ is a weak solution if $\fall \p\in C^{\infty}_{c}(\R^{n})$ we have
\begin{align*}\sums{}{\abs{\al}\leq k}\parens{-1}^{\abs{\al}}\inner{u}{\pd^\al(a_\al\p)}=\inner{b}{\p}\end{align*}
Of course we observe that every solution is a weak solution since we can integrate only within the support of $\p$ which is compact so regularity of solution implies boundedness hence integrable.
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