infimum over all u orthogonal to the . Applications The Dirichlet Laplacian may arise from various problems of mathematical physics; it may refer to modes of at idealized drum, small waves at the surface of an idealized pool, as well as to a mode of an idealized optical fiber in the paraxial approximation. The last application is most practical in connection to the double-clad fibers; 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 shape 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; the pressure integrated over the surface gives the force. 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 of the force of pressure should be also zero. However, there exists a formal proof, which does not refer to the analogy with the physical system. See also Rayleigh–Faber–Krahn inequality Notes References . . Differential operators Partial differential equations Spectral theory