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Difference between revisions of "Strong solution"

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''of a differential equation
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''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>
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$$Lu\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha u=f\label{*}\tag{*}$$
  
in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906002.png" />''
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''in a domain $D$''
  
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
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A locally integrable function $u$ that has locally integrable generalized derivatives of all orders $\leq m$ (cf. [[Generalized derivative|Generalized derivative]]), and satisfies \eqref{*} almost-everywhere in $D$.
  
almost-everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090600/s0906005.png" />.
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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 $L_2$.
  
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
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In the case of an elliptic equation \eqref{*} both notions of a strong solution coincide.
 
 
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.
 
  
  

Latest revision as of 15:58, 14 February 2020

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.


Comments

References

[a1] J. Chazarain, A. Piriou, "Introduction à la théorie des équations aux dérivées partielles linéaires" , Gauthier-Villars (1981) pp. 223 MR0598467 Zbl 0446.35001
How to Cite This Entry:
Strong solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_solution&oldid=28272
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article