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
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
the solution of the Cauchy problem
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 ). The analogous question for was studied in , 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 ). For the latter systems necessary and sufficient conditions of algebraic character have been obtained that ensure the presence of a lacuna.
|||J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952)|
|||M. Mathisson, "Le problème de M. Hadamard rélatif à la diffusion des ondes" Acta Math. , 71 : 3–4 (1939) pp. 249–282|
|||I.G. Petrovskii, "On the diffusion of waves and the lacunas for hyperbolic equations" Mat. Sb. , 17 (1945) pp. 289–370 (In Russian)|
|||R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)|
Further research on lacunae for second-order equations was done by K.L. Stellmacher [a1], R.G. Mclenaghan [a2] and B. Ørsted [a3]. Subsequent to the work  of I.G. Petrovskii, deep investigations were made for the higher-order case by M.F. Atiyah, R. Bott and L. Gårding ; for variable coefficients see also [a5].
|[a1]||K.L. Stellmacher, "Ein Beispiel einer Huyghensschen Differentialgleichung" Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. , 10 (1953) pp. 133–138|
|[a2]||R.G. Mclenaghan, "An explicit determination of the empty space-times on which the wave equation satisfies Huygens' principle" Proc. Cambridge Philos. Soc. , 65 (1969) pp. 139–155|
|[a3]||B. Ørsted, "The conformal invariance of Huygens' principle" J. Diff. Geom. , 16 (1981) pp. 1–9|
|[a4a]||M.F. Atiyah, R. Bott, L. Gårding, "Lacunas for hyperbolic differential operations with constant coefficients I" Acta Math. , 124 (1970) pp. 109–189|
|[a4b]||M.F. Atiyah, R. Bot, L. Gårding, "Lacunas for hyperbolic differential operations with constant coefficients II" Acta Math. , 131 (1973) pp. 145–206|
|[a5]||L. Gårding, "Sharp fronts of paired oscillatory integrals" Publ. Res. Inst. Math. Sci. Kyoto Univ. , 12. Suppl. (1977) pp. 53–68|
Lacuna. A.P. Soldatov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lacuna&oldid=15246