View source for Strong solution

{{TEX|done}}
''of a differential equation''

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

''in a domain $D$''

A locally integrable function $u$ that has locally integrable generalized derivatives of all orders $\leq m$ (cf. [[Generalized derivative|Generalized derivative]]), and satisfies \ref{*} 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 \ref{*} 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 \ref{*} both notions of a strong solution coincide.



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Chazarain,   A. Piriou,   "Introduction à la théorie des équations aux dérivées partielles linéaires" , Gauthier-Villars  (1981)  pp. 223  {{MR|0598467}} {{ZBL|0446.35001}} </TD></TR></table>