View source for Strong solution

''of a differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906001.png" /></td> </tr></table>

in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906002.png" />''

A locally integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906003.png" /> that has locally integrable generalized derivatives of all orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906004.png" /> (cf. [[Generalized derivative|Generalized derivative]]), and satisfies

almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906005.png" />.

The notion of a  "strong solution"  can also be introduced as follows. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906006.png" /> is called a strong solution of

if there are sequences of smooth (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906007.png" />) functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906009.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060012.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060013.png" />, where the convergence is taken in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060014.png" /> for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060015.png" />. In these definitions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060016.png" /> can be replaced by the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060017.png" /> of functions whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060018.png" />-th powers are locally integrable. The class most often used is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s09060019.png" />.

In the case of an elliptic equation

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</TD></TR></table>