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''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 $\Omega$ 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]]

\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|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,

\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|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|linear operator]] in $L _ { 2 } ( \Omega )$ with a discrete spectrum (cf. also [[Spectrum of an operator|Spectrum of an operator]]). Its eigenvalues $0 &lt; \lambda _ { 1 } \leq \lambda _ { 2 } \leq \ldots$ (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 [[#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====
<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>