# Strong solution

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of a differential equation

$$Lu\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha u=f\label{*}\tag{*}$$

in a domain $D$

A locally integrable function $u$ that has locally integrable generalized derivatives of all orders $\leq m$ (cf. Generalized derivative), and satisfies \eqref{*} almost-everywhere in $D$.

The notion of a "strong solution" can also be introduced as follows. A function $u$ is called a strong solution of \eqref{*} if there are sequences of smooth (for example, $C^\infty$) functions $\{u_n\}$, $\{f_n\}$ such that $u_n\to u$, $f_n\to f$ and $Lu_n=f_n$ for each $n$, where the convergence is taken in $L_1(K)$ for any compact set $K\subseteq D$. In these definitions, $L_1$ can be replaced by the class $L_p$ of functions whose $p$-th powers are locally integrable. The class most often used is $L_2$.

In the case of an elliptic equation \eqref{*} both notions of a strong solution coincide.