View source for Dirichlet variational problem

The problem of finding the minimum of the [[Dirichlet integral|Dirichlet integral]]

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under specified boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329502.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329503.png" /> is defined on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329504.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329505.png" />-dimensional region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329506.png" />. The solution of this problem is also the solution of the first boundary value problem for the Laplace equation:

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Dirichlet's variational problem is the first problem concerning minimization of a functional to which the solution of a boundary value problem for a partial differential equation has been reduced.

It is natural to consider Dirichlet's variational problem in the class of functions with generalized first square-summable derivatives. If the region is bounded, this set of functions coincides with the [[Sobolev space|Sobolev space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329508.png" />, and is therefore complete in the appropriate metric. Moreover, each function in this space has boundary values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d0329509.png" /> in the sense of almost-everywhere convergence which, if the boundary is sufficiently smooth, coincides with the boundary values in the sense of convergence in the mean or in the sense of the limit of the boundary values of functions continuous in the closed region which approximate the given function in the metric of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295011.png" /> is a bounded region and if there exists at least one function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295012.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295014.png" /> (such functions are known as permissible), the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295015.png" /> of Dirichlet's variational problem exists and is unique. This solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295016.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295017.png" /> (cf. [[Dirichlet principle|Dirichlet principle]]). If the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295019.png" /> is smooth, then, for the class of permissible functions to be non-empty it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295020.png" />. The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032950/d03295021.png" /> of Dirichlet's variational problem can be found by a direct variational method. These results are generalized to both the case of quadratic elliptic functionals containing higher-order derivatives and the case of unbounded regions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "On Dirichlet's problem for the circle and half-space"  ''Mat. Sb.'' , '''35''' :  2  (1954)  pp. 247–266  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.D. Kudryavtsev,   ''Trudy Mat. Inst. Steklov.'' , '''55'''  (1959)  pp. 1–181</TD></TR></table>