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Difference between revisions of "Potential operator"

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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741301.png" /> of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741302.png" /> into the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741303.png" /> that is the gradient of some functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741304.png" />, i.e. is such that
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<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/p074130/p0741305.png" /></td> </tr></table>
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For instance, any bounded [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741306.png" /> defined on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074130/p0741307.png" /> is potential:
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A mapping  $  A $
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of a [[Banach space|Banach space]] $  X $
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into the dual space $  X  ^ {*} $
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that is the gradient of some functional  $  f \in X  ^ {*} $,
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i.e. is such that
  
<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/p074130/p0741308.png" /></td> </tr></table>
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$$
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\langle  A x , h \rangle  = \lim\limits _ { t\rightarrow } 0 \
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\frac{f ( x + t h ) - f ( x) }{t}
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.
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$$
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For instance, any bounded [[Self-adjoint operator|self-adjoint operator]]  $  A $
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defined on a Hilbert space  $  H $
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is potential:
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$$
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Ax  =   \mathop{\rm grad}  \left \{
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\frac{1}{2}
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\langle  A x , x \rangle \right \} ,\ \
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x \in H .
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$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Vainberg,  "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Gajewski,  K. Gröger,  K. Zacharias,  "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Vainberg,  "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Gajewski,  K. Gröger,  K. Zacharias,  "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag  (1974)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


A mapping $ A $ of a Banach space $ X $ into the dual space $ X ^ {*} $ that is the gradient of some functional $ f \in X ^ {*} $, i.e. is such that

$$ \langle A x , h \rangle = \lim\limits _ { t\rightarrow } 0 \ \frac{f ( x + t h ) - f ( x) }{t} . $$

For instance, any bounded self-adjoint operator $ A $ defined on a Hilbert space $ H $ is potential:

$$ Ax = \mathop{\rm grad} \left \{ \frac{1}{2} \langle A x , x \rangle \right \} ,\ \ x \in H . $$

References

[1] M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian)
[2] H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974)
How to Cite This Entry:
Potential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_operator&oldid=17586
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article