Example

Let $n\in \mathbf{N}$. Let U be an open set in $\mathbf{R}^n$. Endow U with Lebesgue measure, denoted by $\mu$. Let Δ be the scalar differential operator $-\sum^n_{i=1}\partial_i^2$ on U. The partial operator $\Delta_-$ with domain $\mathscr{D}(U)$ defined by $\varphi \mapsto \Delta (\varphi )$ is closable (Proposition 13 of IV, p. 242) and symmetric (IV, p. 243). It is positive, since for every $\varphi \in \mathscr{D}$(U), we have

We denote by $\Delta_D$ the Friedrichs extension of the positive symmetric partial operator $\Delta_-($IV, p. 295, Example); it is a Laplacian on U, called the Dirichlet Laplacian on U.

Let $q$ be the positive partial form associated with $\Delta_D$. The domain of $q$ is the Hilbert space completion of $\mathscr{D}(U)$ for the positive Hermitian form defined by

for every $(\varphi_1, \varphi_2)\in \mathscr{D}(U)\times \mathscr{D}(U)$. In other words, the domain of $q$ is the Sobolev space $H^1_0(U)$ (No.$^o14$ of IV, p. 221).

*Suppose that U is bounded. The canonical injection of $H^1_0(U)$ into $L^2(U)$ is compact; the Dirichlet Laplacian on U is therefore an operator with compact resolvent (Corollary of Proposition 20). Since the Hilbert space $H^1_0(U)$ is of countable type (Proposition 20 of IV, p. 222), and since the image of $\Delta_D$ is of infinite dimension, there exists an increasing sequence $(\lambda_n)_{n\geqslant 0}$ of real numbers tending to $+\infty$ and an orthonormal basis $(f_n)_{n\in\mathbf{N}}$ of $L^2$(U), whose elements belong to the domain of $\Delta_D$, such that $\Delta_D(f_n) =\lambda_nf_n$ for every $n\in \mathbf{N}$. One can prove ("Weyl's law") that as T tends to $+\infty$, one has

$\lambda_n\leqslant T$

where $c_n=\pi^{n/2}/\Gamma (1 +n/2)$ is the volume of the unit ball in $\mathbf{R}^n$ (INT, V, p. 101, § 8, n$^o7$) and $m >0$ is the Lebesgue measure of U.*