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''Dirichlet–Laplace operator''
 
''Dirichlet–Laplace operator''
  
In a broad sense, a restriction of the [[Laplace operator|Laplace operator]] to the space of functions satisfying (in some sense) homogeneous [[Dirichlet boundary conditions|Dirichlet boundary conditions]]. For an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201901.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201902.png" />, the Dirichlet Laplacian is usually defined via the Friedrichs extension procedure. Namely, first consider the (negative) Laplace operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201903.png" /> defined on the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201904.png" /> of all infinitely smooth functions with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201905.png" />. This is a symmetric operator, and the associated quadratic form (with the same domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201906.png" />) is given by the [[Dirichlet integral|Dirichlet integral]]
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In a broad sense, a restriction of the [[Laplace operator|Laplace operator]] to the space of functions satisfying (in some sense) homogeneous [[Dirichlet boundary conditions|Dirichlet boundary conditions]]. For an open set $\Omega$ in ${\bf R} ^ { n }$, the Dirichlet Laplacian is usually defined via the Friedrichs extension procedure. Namely, first consider the (negative) Laplace operator $- \Delta$ defined on the subspace $C _ { 0 } ^ { \infty } ( \Omega ) \subset L _ { 2 } ( \Omega )$ of all infinitely smooth functions with compact support in $\Omega$. This is a symmetric operator, and the associated quadratic form (with the same domain $C _ { 0 } ^ { \infty } ( \Omega )$) is given by the [[Dirichlet integral|Dirichlet integral]]
  
<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/d/d120/d120190/d1201907.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} E ( f ) = \int _ { \Omega } | \nabla f | ^ { 2 } d x. \end{equation}
  
Then the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d1201908.png" /> is closeable with respect to the norm
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Then the form $E$ is closeable with respect to the norm
  
<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/d/d120/d120190/d1201909.png" /></td> </tr></table>
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\begin{equation*} \left( E ( f ) + \| f \| _ { L _ { 2 } ( \Omega ) } \right) ^ { 1 / 2 }. \end{equation*}
  
The domain of its closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019010.png" /> is the [[Sobolev space|Sobolev space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019011.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019012.png" /> (given again by the right-hand side of (a1)) is the quadratic form of a non-negative [[Self-adjoint operator|self-adjoint operator]] (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019013.png" />); moreover,
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The domain of its closure $\tilde { E }$ is the [[Sobolev space|Sobolev space]] $H _ { 0 } ^ { 1 } ( \Omega ) = W _ { 0 } ^ { 1,2 } ( \Omega )$. Then $\tilde { E }$ (given again by the right-hand side of (a1)) is the quadratic form of a non-negative [[Self-adjoint operator|self-adjoint operator]] (denoted by $- \Delta _ { \operatorname{Dir} }$); moreover,
  
<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/d/d120/d120190/d12019014.png" /></td> </tr></table>
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\begin{equation*} \operatorname { Dom } \left( ( - \Delta _ { \text{Dir} } ) ^ { 1 / 2 } \right) = \operatorname { Dom } ( \tilde{E} ) = H _ { 0 } ^ { 1 } ( \Omega ). \end{equation*}
  
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019015.png" /> (sometimes taken with the minus sign) is called the Dirichlet Laplacian (in the weak sense).
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The operator $\Delta_{\operatorname{ Dir}}$ (sometimes taken with the minus sign) is called the Dirichlet Laplacian (in the weak sense).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019016.png" /> is bounded domain with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019017.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019018.png" />, then
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If $\Omega$ is bounded domain with boundary $\partial \Omega$ of class $C ^ { 2 }$, then
  
<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/d/d120/d120190/d12019019.png" /></td> </tr></table>
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\begin{equation*} \operatorname{Dom} ( - \Delta_{\text{ Dir}} ) = H _ { 0 } ^ { 1 } ( \Omega ) \bigcap H ^ { 2 } ( \Omega ). \end{equation*}
  
 
The Dirichlet Laplacian for a compact [[Riemannian manifold|Riemannian manifold]] with boundary is defined similarly.
 
The Dirichlet Laplacian for a compact [[Riemannian manifold|Riemannian manifold]] with boundary is defined similarly.
  
For a bounded open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019022.png" /> is a positive unbounded [[Linear operator|linear operator]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019023.png" /> with a discrete spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]). Its eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019024.png" /> (written in increasing order with account of multiplicity) can be found using the Rayleigh–Ritz variational formula (or max-min formula)
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For a bounded open set $\Omega$ in ${\bf R} ^ { n }$, $- \Delta _ { \operatorname{Dir} }$ is a positive unbounded [[Linear operator|linear operator]] in $L _ { 2 } ( \Omega )$ with a discrete spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]). Its eigenvalues $0 < \lambda _ { 1 } \leq \lambda _ { 2 } \leq \ldots$ (written in increasing order with account of multiplicity) can be found using the Rayleigh–Ritz variational formula (or max-min formula)
  
<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/d/d120/d120190/d12019025.png" /></td> </tr></table>
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\begin{equation*} \lambda _ { n } ( \Omega ) = \operatorname { inf } \{ \lambda ( L ) : L \subseteq C ^ { \infty _0 } ( \Omega ) , \operatorname { dim } ( L ) = n \}, \end{equation*}
  
 
where
 
where
  
<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/d/d120/d120190/d12019026.png" /></td> </tr></table>
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\begin{equation*} \lambda ( L ) = \operatorname { sup } \{ E ( f ) : f \in L , \| f \| _ { L _ { 2 } ( \Omega ) } = 1 \} \end{equation*}
  
for a finite-dimensional linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019028.png" />. It follows from the Rayleigh–Ritz formula that the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019029.png" /> are monotonically decreasing functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120190/d12019030.png" />. See also [[#References|[a3]]] for a survey of the asymptotic behaviour of the eigenvalues of the Dirichlet Laplacian and operators corresponding to other boundary value problems for elliptic differential operators.
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for a finite-dimensional linear subspace $L$ of $C _ { 0 } ^ { \infty } ( \Omega )$. It follows from the Rayleigh–Ritz formula that the eigenvalues $\lambda _ { n }$ are monotonically decreasing functions of $\Omega$. See also [[#References|[a3]]] for a survey of the asymptotic behaviour of the eigenvalues of the Dirichlet Laplacian and operators corresponding to other boundary value problems for elliptic differential operators.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.B. Davies,  "Spectral theory and differential operators" , Cambridge Univ. Press  (1995)  {{MR|1349825}} {{ZBL|0893.47004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.E. Edmunds,  W.D. Ewans,  "Spectral theory and differential operators" , Clarendon Press  (1987)  {{MR|0929030}} {{ZBL|0628.47017}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Yu. Safarov,  D. Vassiliev,  "The asymptotic distribution of eigenvalues of partial differential operators" , ''Transl. Math. Monogr.'' , '''55''' , Amer. Math. Soc.  (1997)  {{MR|1414899}} {{ZBL|0898.35003}} {{ZBL|0870.35003}} </TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  E.B. Davies,  "Spectral theory and differential operators" , Cambridge Univ. Press  (1995)  {{MR|1349825}} {{ZBL|0893.47004}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D.E. Edmunds,  W.D. Ewans,  "Spectral theory and differential operators" , Clarendon Press  (1987)  {{MR|0929030}} {{ZBL|0628.47017}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top">  Yu. Safarov,  D. Vassiliev,  "The asymptotic distribution of eigenvalues of partial differential operators" , ''Transl. Math. Monogr.'' , '''55''' , Amer. Math. Soc.  (1997)  {{MR|1414899}} {{ZBL|0898.35003}} {{ZBL|0870.35003}} </td></tr>
 +
</table>

Latest revision as of 19:34, 7 February 2024

Dirichlet–Laplace operator

In a broad sense, a restriction of the Laplace operator to the space of functions satisfying (in some sense) homogeneous Dirichlet boundary conditions. For an open set $\Omega$ in ${\bf R} ^ { n }$, the Dirichlet Laplacian is usually defined via the Friedrichs extension procedure. Namely, first consider the (negative) Laplace operator $- \Delta$ defined on the subspace $C _ { 0 } ^ { \infty } ( \Omega ) \subset L _ { 2 } ( \Omega )$ of all infinitely smooth functions with compact support in $\Omega$. This is a symmetric operator, and the associated quadratic form (with the same domain $C _ { 0 } ^ { \infty } ( \Omega )$) is given by the Dirichlet integral

\begin{equation} \tag{a1} E ( f ) = \int _ { \Omega } | \nabla f | ^ { 2 } d x. \end{equation}

Then the form $E$ is closeable with respect to the norm

\begin{equation*} \left( E ( f ) + \| f \| _ { L _ { 2 } ( \Omega ) } \right) ^ { 1 / 2 }. \end{equation*}

The domain of its closure $\tilde { E }$ is the Sobolev space $H _ { 0 } ^ { 1 } ( \Omega ) = W _ { 0 } ^ { 1,2 } ( \Omega )$. Then $\tilde { E }$ (given again by the right-hand side of (a1)) is the quadratic form of a non-negative self-adjoint operator (denoted by $- \Delta _ { \operatorname{Dir} }$); moreover,

\begin{equation*} \operatorname { Dom } \left( ( - \Delta _ { \text{Dir} } ) ^ { 1 / 2 } \right) = \operatorname { Dom } ( \tilde{E} ) = H _ { 0 } ^ { 1 } ( \Omega ). \end{equation*}

The operator $\Delta_{\operatorname{ Dir}}$ (sometimes taken with the minus sign) is called the Dirichlet Laplacian (in the weak sense).

If $\Omega$ is bounded domain with boundary $\partial \Omega$ of class $C ^ { 2 }$, then

\begin{equation*} \operatorname{Dom} ( - \Delta_{\text{ Dir}} ) = H _ { 0 } ^ { 1 } ( \Omega ) \bigcap H ^ { 2 } ( \Omega ). \end{equation*}

The Dirichlet Laplacian for a compact Riemannian manifold with boundary is defined similarly.

For a bounded open set $\Omega$ in ${\bf R} ^ { n }$, $- \Delta _ { \operatorname{Dir} }$ is a positive unbounded linear operator in $L _ { 2 } ( \Omega )$ with a discrete spectrum (cf. also Spectrum of an operator). Its eigenvalues $0 < \lambda _ { 1 } \leq \lambda _ { 2 } \leq \ldots$ (written in increasing order with account of multiplicity) can be found using the Rayleigh–Ritz variational formula (or max-min formula)

\begin{equation*} \lambda _ { n } ( \Omega ) = \operatorname { inf } \{ \lambda ( L ) : L \subseteq C ^ { \infty _0 } ( \Omega ) , \operatorname { dim } ( L ) = n \}, \end{equation*}

where

\begin{equation*} \lambda ( L ) = \operatorname { sup } \{ E ( f ) : f \in L , \| f \| _ { L _ { 2 } ( \Omega ) } = 1 \} \end{equation*}

for a finite-dimensional linear subspace $L$ of $C _ { 0 } ^ { \infty } ( \Omega )$. It follows from the Rayleigh–Ritz formula that the eigenvalues $\lambda _ { n }$ are monotonically decreasing functions of $\Omega$. See also [a3] for a survey of the asymptotic behaviour of the eigenvalues of the Dirichlet Laplacian and operators corresponding to other boundary value problems for elliptic differential operators.

References

[a1] E.B. Davies, "Spectral theory and differential operators" , Cambridge Univ. Press (1995) MR1349825 Zbl 0893.47004
[a2] D.E. Edmunds, W.D. Ewans, "Spectral theory and differential operators" , Clarendon Press (1987) MR0929030 Zbl 0628.47017
[a3] Yu. Safarov, D. Vassiliev, "The asymptotic distribution of eigenvalues of partial differential operators" , Transl. Math. Monogr. , 55 , Amer. Math. Soc. (1997) MR1414899 Zbl 0898.35003 Zbl 0870.35003
How to Cite This Entry:
Dirichlet Laplacian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_Laplacian&oldid=28180
This article was adapted from an original article by M. Levitin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article