View source for Dirichlet Laplacian

''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]]

<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>

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

<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>

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,

<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>

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).

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

<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>

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)

<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>

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>

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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.B. Davies,   "Spectral theory and differential operators" , Cambridge Univ. Press  (1995)</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)</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)</TD></TR></table>