Consider a bounded domain with a piecewise smooth boundary . is a Dirichlet eigenvalue of if there exists a function (a Dirichlet eigenfunction) satisfying the following Dirichlet boundary value problem (cf. also Dirichlet boundary conditions):
where is the Laplace operator (i.e., ). Dirichlet eigenvalues (with ) 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 is bounded and the boundary is sufficiently regular, the Dirichlet Laplacian has a discrete spectrum of infinitely many positive eigenvalues with no finite accumulation point [a15]:
( as ).
The Dirichlet eigenvalues are characterized by the max-min principle [a4]:
where the is taken over all orthogonal to , and the is taken over all choices of . For simply-connected domains it follows from the max-min principle (a4) that is non-degenerate and the corresponding eigenfunction is positive in the interior of . For higher values of the nodal lines of the th eigenfunction divide into no more than 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 is a bounded, smooth, convex domain, the nodal line of always meets ).
where and are, respectively, the volumes of and of the unit ball in .
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
and conjectured the same bound for any bounded domain in (here is the area of the domain). Pólya's conjecture in dimensions is equivalent to saying that the Weyl asymptotics of , (a5), is a lower bound for , i.e.,
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]
proven using the asymptotic behaviour of the heat kernel of (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).
Dirichlet eigenvalues are completely characterized by the geometry of the domain . The inverse problem, i.e., up to what extent the geometry of can be recovered from the knowledge of , was posed by M. Kac in [a8]. If , for example, and is smooth (in particular does not have corners), then the distribution function behaves as
as , where is the area, the perimeter and the number of holes of , 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 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 . A typical such comparison is the Rayleigh–Faber–Krahn inequality. Another inequality along these lines is the following: If is a simply connected domain in and is the radius of the largest disc contained in , then there is a universal constant such that
(as of 2000, the best, not yet optimal, constant in (a10) is ; 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 : If and are domains in , and one sets , then for all . 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 (see, e.g., [a16], p. 263), and also for the Laplace–Beltrami operator defined on domains in Riemannian manifolds (see, e.g., [a5]).
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Dirichlet eigenvalue. Rafael D. Benguria (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dirichlet_eigenvalue&oldid=14188