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Difference between revisions of "Principal fundamental solution"

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A [[Fundamental solution|fundamental solution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747001.png" />, defined throughout the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747002.png" />, of a second-order elliptic equation
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A [[Fundamental solution|fundamental solution]] $G(x,y)$, defined throughout the space $E^n$, of a second-order elliptic 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/p/p074/p074700/p0747003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$Au=\sum_{i,k=1}^na_{ik}\frac{\partial^2u}{\partial x_i\partial x_k}+\sum_{i=1}^nb_i(x)\frac{\partial u}{\partial x_i}+c(x)u=0\tag{*}$$
  
 
that satisfies the conditions
 
that satisfies the conditions
  
<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/p/p074/p074700/p0747004.png" /></td> </tr></table>
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$$G(x,y)=o(e^{-a|x-y|}),\quad\frac{\partial G}{\partial x_i}=o(e^{-a|x-y|})$$
  
for certain positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747006.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747007.png" />.
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for certain positive constants $a$ and $R$ if $|x-y|>R$.
  
If the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p0747009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p07470010.png" /> satisfy a [[Hölder condition|Hölder condition]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p07470011.png" /> and if the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p07470012.png" /> is satisfied for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p07470013.png" />, then a principal fundamental solution exists. If the coefficients of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p07470014.png" /> are defined in a certain bounded domain with smooth boundary, then they can be extended to the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074700/p07470015.png" /> so that a principal fundamental solution will exist for the extended operator.
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If the coefficients $a_{ik}(x)$, $b_i(x)$ and $c(x)$ satisfy a [[Hölder condition|Hölder condition]] on $E^n$ and if the inequality $c(x)<-\gamma$ is satisfied for some $\gamma>0$, then a principal fundamental solution exists. If the coefficients of the operator $A$ are defined in a certain bounded domain with smooth boundary, then they can be extended to the entire space $E^n$ so that a principal fundamental solution will exist for the extended operator.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>

Latest revision as of 08:35, 31 October 2014

A fundamental solution $G(x,y)$, defined throughout the space $E^n$, of a second-order elliptic equation

$$Au=\sum_{i,k=1}^na_{ik}\frac{\partial^2u}{\partial x_i\partial x_k}+\sum_{i=1}^nb_i(x)\frac{\partial u}{\partial x_i}+c(x)u=0\tag{*}$$

that satisfies the conditions

$$G(x,y)=o(e^{-a|x-y|}),\quad\frac{\partial G}{\partial x_i}=o(e^{-a|x-y|})$$

for certain positive constants $a$ and $R$ if $|x-y|>R$.

If the coefficients $a_{ik}(x)$, $b_i(x)$ and $c(x)$ satisfy a Hölder condition on $E^n$ and if the inequality $c(x)<-\gamma$ is satisfied for some $\gamma>0$, then a principal fundamental solution exists. If the coefficients of the operator $A$ are defined in a certain bounded domain with smooth boundary, then they can be extended to the entire space $E^n$ so that a principal fundamental solution will exist for the extended operator.

References

[1] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)
How to Cite This Entry:
Principal fundamental solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_fundamental_solution&oldid=34113
This article was adapted from an original article by Sh.A. Alimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article