Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Dirichlet eigenvalues
are the fundamental modes of vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...
of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory...
is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function u
≠ 0 and eigenvalue λ
Here Δ is the Laplacian, which is given in xy
The boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
is, of course, the Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
for the Helmholtz equation
The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation\nabla^2 A + k^2 A = 0where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.-Motivation and uses:...
, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator Δ appearing in is often known as the Dirichlet Laplacian
when it is considered as accepting only functions u
satisfying the Dirichlet boundary condition. More generally, in spectral geometry
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other...
one considers on a manifold with boundary Ω. Then Δ is taken to be the Laplace-Beltrami operator
In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami...
, also with Dirichlet boundary conditions.
It can be shown, using the spectral theorem for compact self-adjoint operators that the eigenspaces are finite-dimensional and that the Dirichlet eigenvalues λ are real, positive, and have no limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
. Thus they can be arranged in increasing order:
where each eigenvalue is counted according to its geometric multiplicity. The eigenspaces are orthogonal in the space of square-integrable function
In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...
s, and consist of smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s. In fact, the Dirichlet Laplacian has a continuous extension to an operator from the Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
. This operator is invertible, and its inverse is compact and self-adjoint so that the usual spectral theorem can be applied to obtain the eigenspaces of Δ and the reciprocals 1/λ of its eigenvalues.
One of the primary tools in the study of the Dirichlet eigenvalues is the max-min principle: the first eigenvalue λ1
minimizes the Dirichlet energy
In mathematics, the Dirichlet's energy is a numerical measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space...
. To wit,
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
is taken over all u
of compact support that do not vanish identically in Ω. By a density argument, this infimum agrees with that taken over nonzero
. Moreover, using results from the calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
analogous to the Lax–Milgram theorem, one can show that a minimizer exists in
. More generally, one has
where the supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
is taken over all (k
and the infimum over all u
orthogonal to the φi
The Dirichlet Laplacian may arise from various problems of mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
it may refer to modes of at idealized drum
The drum is a member of the percussion group of musical instruments, which is technically classified as the membranophones. Drums consist of at least one membrane, called a drumhead or drum skin, that is stretched over a shell and struck, either directly with the player's hands, or with a...
, small waves at the surface of an idealized pool
A pond is a body of standing water, either natural or man-made, that is usually smaller than a lake. A wide variety of man-made bodies of water are classified as ponds, including water gardens, water features and koi ponds; all designed for aesthetic ornamentation as landscape or architectural...
as well as to a mode of an idealized optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...
in the paraxial approximation
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system ....
The last application is most practical in connection to the double-clad fiber
Double-clad fiber is a class of optical fiber with a structure consisting of three layers of optical material instead of the usual two. The inner-most layer is called the core. It is surrounded by the inner cladding, which is surrounded by the outer cladding...
in such fibers, it is important, that most of modes of the fill the domain uniformly,
or the most of rays cross the core. The poorest chape seems to be the circularly-symmetric domain
The modes of pump should not avoid the active core used in double-clad fiber amplifiers.
The spiral-shaped domain happens to be especially efficient for such an application due to the
boundary behavior of modes of Dirichlet laplacian
The theorem about boundary behavior of the Dirichlet Laplacian if analogy of the property of rays in geometrical optics (Fig.1);
the angular momentum of a ray (green) increases at each reflection from the spiral part of the boundary (blue), until the ray hits the chunk (red); all rays (except those parallel to the optical axis) unavoidly visit the region in vicinity of the chunk to frop the excess of the
angular momentum. Similarly, all the modes of the Dirichlet Laplacian have non-zero values in vicinity of the chunk. The normal component of the derivative
of the mode at the boundary can be interpreted as pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
; the pressure integrated over the surface gives the force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
. As the mode is steady-state
solution of the propagation equation (with trivial dependence of the longitudinal coordinate), the total force should be zero.
Similarly, the angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of the force of pressure should be also zero. However, there exist the formal proof, which
does not refer to the analogy with physical system.