View source for Vladimirov variational principle

A variational principle for the stationary single-velocity homogeneous transport equation (cf. [[Transport equations, numerical methods|Transport equations, numerical methods]])

<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/v/v096/v096820/v0968201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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with boundary condition

<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/v/v096/v096820/v0968203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968204.png" /> is the boundary of a convex bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968205.png" />. If the scatter indicatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968206.png" /> is an even function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968207.png" />, transition to the new unknown function

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reduces the problem (1), (2) to self-adjoint form. In the problem thus obtained, Vladimirov's variational principle for the smallest eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v0968209.png" /> states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v09682010.png" /> is the minimum of the functional

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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/v/v096/v096820/v09682012.png" /></td> </tr></table>

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on the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096820/v09682014.png" /> which satisfy the condition

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The corresponding (non-negative) eigenfunction realizes the minimum of the functional [[#References|[3]]]. In this variational principle the respective boundary conditions are natural. Variational principles for higher eigenvalues and for the non-homogeneous problem are formulated in a similar manner.

The principle was first obtained by V.S. Vladimirov [[#References|[1]]], and it yielded the optimum boundary conditions in the method of spherical harmonics (cf. [[Spherical harmonics, method of|Spherical harmonics, method of]]). Vladimirov's variational principle, in conjunction with finite difference methods, is extensively employed in numerical computations of neutron physics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Mathematical problems in the theory of single-velocity particle transfer" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.I. Marchuk,   "Methods of calculation of nuclear reactors" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Davison,   "Neutron transport theory" , Oxford Univ. Press  (1957)</TD></TR></table>



====Comments====
Cf. also [[Vladimirov method|Vladimirov method]].