View source for Dirichlet integral

A functional connected with the solution of the [[Dirichlet problem|Dirichlet problem]] for the Laplace equation by the variational method. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328701.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328702.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328703.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328704.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328705.png" /> and let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328706.png" /> (cf. [[Sobolev space|Sobolev space]]). The Dirichlet integral for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328707.png" /> is the expression

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For a certain given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d0328709.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287010.png" /> one considers the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287011.png" /> of functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287012.png" /> which satisfy the boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287013.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287014.png" /> is non-empty, there exists a unique function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287015.png" /> for which

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and this function is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287017.png" />. The converse theorem is also true: If a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287018.png" /> belongs to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287020.png" /> is attained on it. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287021.png" /> is a generalized solution from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287022.png" /> of the Dirichlet problem for the Laplace equation. However, not for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287023.png" /> it is possible to find a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287024.png" />. There exists even continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287025.png" /> for which the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287026.png" /> is empty, i.e. the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287027.png" /> contains no functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287028.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287029.png" />. The classical solution of the Dirichlet problem for the Laplace equation with such boundary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287030.png" /> cannot have a finite Dirichlet integral and is not a generalized solution from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287031.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Mikhailov,   "Partial differential equations" , MIR  (1978)  (Translated from Russian)</TD></TR></table>



====Comments====
The restriction of a function (distribution) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287032.png" /> to a set (in this case the boundary) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287033.png" /> is also called the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287035.png" /> in this setting.

See [[#References|[a1]]] for a well-known additional reference. Note that the Hilbert space obtained by completion of the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287036.png" />-functions with compact support with respect to the scalar product

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can be continuously imbedded into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032870/d03287038.png" />. This observation leads to the introduction of the axiomatic theory of Dirichlet spaces, explaining larger parts of classical potential theory (see, e.g., [[#References|[a2]]] or [[#References|[a3]]], and [[Potential theory|Potential theory]]).

See also [[Dirichlet principle|Dirichlet principle]]; [[Dirichlet variational problem|Dirichlet variational problem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Deny,   "Méthodes Hilbertiennes et théorie du potential"  M. Brelot (ed.)  H. Bauer (ed.)  J.-M. Bony (ed.)  J. Deny (ed.)  G. Mokobodzki (ed.) , ''Potential theory (CIME, Stresa, 1969)'' , Cremonese  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Fukushima,   "Dirichlet forms and Markov processes" , North-Holland  (1980)</TD></TR></table>