Dirichlet Laplacian

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