Dirichlet eigenvalue

Consider a bounded domain $\Omega \subset \mathbf{R} ^ { n }$ with a piecewise smooth boundary $\partial \Omega$. $\lambda$ is a Dirichlet eigenvalue of $\Omega$ if there exists a function $u \in C ^ { 2 } ( \Omega ) \cap C ^ { 0 } ( \overline { \Omega } )$ (a Dirichlet eigenfunction) satisfying the following Dirichlet boundary value problem (cf. also Dirichlet boundary conditions):

\begin{equation} \tag{a1} - \Delta u = \lambda u \text { in } \Omega, \end{equation}

\begin{equation} \tag{a2} u = 0 \text { in } \partial \Omega, \end{equation}

where $\Delta$ is the Laplace operator (i.e., $\Delta = \sum _ { i = 1 } ^ { n } \partial ^ { 2 } / \partial x _ { i } ^ { 2 }$). Dirichlet eigenvalues (with $n = 2$) were introduced in the study of the vibrations of the clamped membrane in the nineteenth century. In fact, they are proportional to the square of the eigenfrequencies of the membrane with fixed boundary. See [a9] for a review and historical remarks. Provided $\Omega$ is bounded and the boundary $\partial \Omega$ is sufficiently regular, the Dirichlet Laplacian has a discrete spectrum of infinitely many positive eigenvalues with no finite accumulation point [a15]:

\begin{equation} \tag{a3} 0 < \lambda _ { 1 } ( \Omega ) \leq \lambda _ { 2 } ( \Omega ) \leq \dots \end{equation}

($\lambda _ { k } \rightarrow \infty$ as $ k \rightarrow \infty$).

The Dirichlet eigenvalues are characterized by the max-min principle [a4]:

\begin{equation} \tag{a4} \lambda _ { k } = \operatorname { sup } \operatorname { inf } \frac { \int _ { \Omega } ( \nabla u ) ^ { 2 } d x } { \int _ { \Omega } u ^ { 2 } d x }, \end{equation}

where the $\operatorname{inf}$ is taken over all $u \in H _ { 0 } ^ { 1 } ( \Omega )$ orthogonal to $\varphi _ { 1 } , \dots , \varphi _ { k - 1 } \in H _ { 0 } ^ { 1 } ( \Omega )$, and the $\operatorname {sup}$ is taken over all choices of $\{ \varphi _ { i } \} _ { i = 1 } ^ { k - 1 }$. For simply-connected domains it follows from the max-min principle (a4) that $\lambda _ { 1 } ( \Omega )$ is non-degenerate and the corresponding eigenfunction $u_1$ is positive in the interior of $\Omega$. For higher values of $k$ the nodal lines of the $k$th eigenfunction divide $\Omega$ into no more than $k - 1$ subregions (nodal domains; this is Courant's nodal line theorem [a4]). Along this subject, notice the proof of A.D. Melas [a11] of the nodal line conjecture for plane domains (if $\Omega$ is a bounded, smooth, convex domain, the nodal line of $u_2$ always meets $\partial \Omega$).

Weyl asymptotics.

For large values of $k$, if $\Omega \subset \mathbf{R} ^ { n }$, H. Weyl [a17], [a18] proved

\begin{equation} \tag{a5} \lambda _ { k } \approx \frac { 4 \pi ^ { 2 } k ^ { 2 / n } } { ( C _ { n } | \Omega | ) ^ { 2 / n } }, \end{equation}

where $| \Omega |$ and $C _ { n } = \pi ^ { n / 2 } / \Gamma ( n / 2 + 1 )$ are, respectively, the volumes of $\Omega$ and of the unit ball in ${\bf R} ^ { n }$.

Pólya conjecture.

For any plane-covering domain (i.e., a domain that can be used to tile the plane without gaps, nor overlaps, allowing rotations, translations and reflections of itself), G. Pólya [a13] proved that

\begin{equation} \tag{a6} \lambda _ { k } \geq \frac { 4 \pi k } { A }\; \text { for } k = 1,2 , \ldots, \end{equation}

and conjectured the same bound for any bounded domain in $\mathbf{R} ^ { 2 }$ (here $A$ is the area of the domain). Pólya's conjecture in $n$ dimensions is equivalent to saying that the Weyl asymptotics of $\lambda _ { k }$, (a5), is a lower bound for $\lambda _ { k }$, i.e.,

\begin{equation} \tag{a7} \lambda _ { k } \geq \frac { 4 \pi ^ { 2 } k ^ { 2 / n } } { ( C _ { n } | \Omega | ) ^ { 2 / n } } \text { for } k = 1,2, \dots . \end{equation}

A result analogous to (a6) for the Neumann eigenvalues of tiling domains, with the sign of the equalities reversed, also holds (cf. also Neumann eigenvalue). The best result to date (2000) towards the proof of the Pólya conjecture is the bound [a10]

\begin{equation} \tag{a8} \sum _ { i = 1 } ^ { k } \lambda _ { i } \geq \frac { n } { n + 2 } \frac { 4 \pi ^ { 2 } k ^ { 1 + 2 / n } } { ( C _ { n } | \Omega | ) ^ { 2 / n } } k = 1,2 , \ldots , \end{equation}

proven using the asymptotic behaviour of the heat kernel of $\Omega$ (cf. also Heat equation) and the connection between the heat kernel and the Dirichlet eigenvalues of a domain (see, e.g., [a6] for a review and related results).

Kac problem.

Dirichlet eigenvalues are completely characterized by the geometry of the domain $\Omega$. The inverse problem, i.e., up to what extent the geometry of $\Omega$ can be recovered from the knowledge of $\{ \lambda _ { n } \} _ { n = 1 } ^ { \infty }$, was posed by M. Kac in [a8]. If $n = 2$, for example, and $\partial \Omega$ is smooth (in particular $\partial \Omega$ does not have corners), then the distribution function behaves as

\begin{equation} \tag{a9} \sum _ { k = 1 } ^ { \infty } e ^ { - \lambda _ { k } t } \approx \frac { A } { 4 \pi t } + \frac { L } { 8 \sqrt { \pi } t } + \frac { 1 } { 6 } ( 1 - r ) + O ( t ), \end{equation}

as $t \rightarrow 0$, where $A$ is the area, $L$ the perimeter and $r$ the number of holes of $\Omega$, so at least these features of the domain can be recovered from knowledge of all the eigenvalues (the first term in (a9) is just a consequence of Weyl's asymptotics). However, complete recovery of the geometry is impossible, as was later shown by C. Gordon, D. Web and S. Wolpert, who constructed two isospectral domains in $\mathbf{R} ^ { 2 }$ with different geometries [a7].

Eigenvalues and geometry.

The inverse of the square root of a Dirichlet eigenvalue is a length that may be compared with other characteristic lengths of the domain $\Omega$. A typical such comparison is the Rayleigh–Faber–Krahn inequality. Another inequality along these lines is the following: If $\Omega$ is a simply connected domain in $\mathbf{R} ^ { 2 }$ and $r _ { \Omega }$ is the radius of the largest disc contained in $\Omega$, then there is a universal constant $a$ such that

\begin{equation} \tag{a10} \lambda _ { 1 } ( \Omega ) \geq \frac { a } { r _ { \Omega } ^ { 2 } } \end{equation}

(as of 2000, the best, not yet optimal, constant in (a10) is $a = 0.6197$; see [a2] for details and historical facts). For other isoperimetric inequalities, see, e.g., [a1], [a12], [a14]. In the same vein, one can also compare Dirichlet and Neumann eigenvalues (see Neumann eigenvalue).

Because of the connection between potential theory and Brownian motion, it is possible to use probabilistic methods to find properties of Dirichlet eigenvalues. One such property was found by H. Brascamp and E.H. Lieb [a3] for $\lambda _ { 1 }$: If $\Omega _ { 1 }$ and $\Omega _ { 2 }$ are domains in ${\bf R} ^ { n }$, and one sets $\Omega _ { t } = t \Omega _ { 1 } + ( 1 - t ) \Omega _ { 2 }$, then $\lambda _ { 1 } ( \Omega _ { t } ) \leq t \lambda _ { 1 } ( \Omega _ { 1 } ) + ( 1 - t ) \lambda _ { 2 } ( \Omega _ { 2 } )$ for all $t \in ( 0,1 )$. Another example of the use of probabilistic methods is the proof of (a10) by R. Bañuelos and T. Carroll [a2].

To conclude, note that it is possible to define Dirichlet eigenvalues for much more general domains in ${\bf R} ^ { n }$ (see, e.g., [a16], p. 263), and also for the Laplace–Beltrami operator defined on domains in Riemannian manifolds (see, e.g., [a5]).

References

[a1]	M.S. Ashbaugh, R.D. Benguria, "Isoperimetric inequalities for eigenvalue ratios" , Symp. Math. , 35 , Cambridge Univ. Press (1994) pp. 1–36
[a2]	R. Bañuelos, T. Carroll, "Brownian motion and the fundamental frequency of a drum" Duke Math. J. , 75 (1994) pp. 575–602
[a3]	H. Brascamp, E.H. Lieb, "On extensions of the Brunn–Minkowski and Prékopa–Leindler theorem, including inequalities for log-concave functions, and with an application to the diffusion equation" J. Funct. Anal. , 22 (1976) pp. 366–389
[a4]	R. Courant, D. Hilbert, "Methoden der mathematischen Physik" , I , Springer (1931) (English transl.: Methods of mathematical physics, vol. I., Interscience, 1953)
[a5]	I. Chavel, "Eigenvalues in Riemannian geometry" , Pure Appl. Math. , 115 , Acad. Press (1984)
[a6]	E.B. Davies, "Heat kernels and spectral theory" , Tracts in Math. , 92 , Cambridge Univ. Press (1989)
[a7]	C. Gordon, D. Webb, S. Wolpert, "Isospectral plane domains and surfaces via Riemannian orbifolds" Invent. Math. , 110 (1992) pp. 1–22
[a8]	M. Kac, "Can one hear the shape of a drum?" Amer. Math. Monthly , 73 : 4 (1966) pp. 1–23
[a9]	J.R. Kuttler, V.G. Sigillito, "Eigenvalues of the Laplacian in two dimensions" SIAM Review , 26 (1984) pp. 163–193
[a10]	P. Li, S.T. Yau, "On the Schrödinger equation and the eigenvalue problem" Commun. Math. Phys. , 88 (1983) pp. 309–318
[a11]	A.D. Melas, "On the nodal line of the second eigenfunction of the Laplacian in $\mathbf{R} ^ { 2 }$" J. Diff. Geom. , 35 (1992) pp. 255–263
[a12]	R. Osserman, "Isoperimetric inequalities and eigenvalues of the Laplacian" , Proc. Internat. Congress of Math. Helsinki , Acad. Sci. Fennica (1978) pp. 435–441
[a13]	G. Polya, "On the eigenvalues of vibrating membranes" Proc. London Math. Soc. , 11 : 3 (1961) pp. 419–433
[a14]	G. Polya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Ann. of Math. Stud. , 27 , Princeton Univ. Press (1951)
[a15]	F. Pockels, "Über die partielle Differentialgleichung $\Delta u + k ^ { 2 } u = 0$ und deren Auftreten in die mathematischen Physik" Z. Math. Physik , 37 (1892) pp. 100–105
[a16]	M. Reed, B. Simon, "Methods of modern mathematical physics IV: Analysis of operators" , Acad. Press (1978)
[a17]	H. Weyl, "Ramifications, old and new, of the eigenvalue problem" Bull. Amer. Math. Soc. , 56 (1950) pp. 115–139
[a18]	H. Weyl, "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen" Math. Ann. , 71 (1911) pp. 441–479