# Lacuna

For lacunae in function theory see e.g. Hadamard theorem on gaps; Fabry theorem on gaps; Lacunary power series.

For lacunae in geometry see Group of motions; Lacunary space.

A lacuna in the theory of partial differential equations is a subdomain of the intersection of the interior of the characteristic cone of a linear hyperbolic system

 (1)

with vertex at the point and a plane . This subdomain is defined by the following property: small sufficiently smooth changes of the initial data inside do not affect the value of the solution at the point . In (1) it is assumed that is a linear differential operator of order and that the order of the differentiations in it with respect to does not exceed . A "change inside" means a change in some domain that together with its boundary lies in .

For the wave equation

 (2)

the solution of the Cauchy problem

 (3)

at the point , , is completely determined by the values of the functions and on the sphere for odd and in the ball for even and , hence the domain in the plane is a lacuna for equation (2) for odd . For even and for equation (2) has no lacuna. This agrees with the Huygens principle for solutions of the wave equation.

A perturbation of the initial data (3) in a small spherical neighbourhood of the point leads to a spherical wave with centre at this point, which for odd has outward and inward facing fronts. For the remaining values of the inward facing front of this wave is "diffused" ; this phenomenon is called diffusion of waves. Diffusion of waves is characteristic of all linear second-order hyperbolic equations if the number of space variables is even (see [1]). The analogous question for was studied in [2], where a class of second-order hyperbolic equations was described for which diffusion of waves is absent. The equations of this class are closely connected with the wave equation. For general hyperbolic systems (1) a relation "locally" has been found between the existence of a lacuna for the system (1) and the analogous question for the corresponding system with constant coefficients (see [3]). For the latter systems necessary and sufficient conditions of algebraic character have been obtained that ensure the presence of a lacuna.

#### References

 [1] J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952) [2] M. Mathisson, "Le problème de M. Hadamard rélatif à la diffusion des ondes" Acta Math. , 71 : 3–4 (1939) pp. 249–282 [3] I.G. Petrovskii, "On the diffusion of waves and the lacunas for hyperbolic equations" Mat. Sb. , 17 (1945) pp. 289–370 (In Russian) [4] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)