View source for Weak solution

''of a differential equation

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in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972402.png" />''

A locally integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972403.png" /> satisfying the equation

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for all smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972405.png" /> (say, of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972406.png" />) with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972407.png" />. Here, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972408.png" /> in

are assumed to be sufficiently smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w0972409.png" /> stands for the formal Lagrange adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w09724010.png" />:

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For example, the generalized derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w09724012.png" /> can be defined as the locally integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w09724013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w09724014.png" /> is a weak solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097240/w09724015.png" />.

In considering weak solutions of , the following problem arises: under what conditions are they strong solutions (cf. [[Strong solution|Strong solution]])? For example, in the case of elliptic equations, every weak solution is strong.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "Some classes of partial differential equations" , Gordon &amp; Breach  (1988)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Agmon,   "Lectures on elliptic boundary value problems" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gilbarg,   N.S. Trudinger,   "Elliptic partial differential equations of second order" , Springer  (1983)</TD></TR></table>