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 $ D $ of the intersection of the interior of the characteristic cone of a linear hyperbolic system

$$ \tag{1 } \frac{\partial ^ {n _ {i} } u _ {i} }{\partial t ^ {n _ {i} } } = \ \sum _ { j= 1} ^ { k } L _ {ij} u _ {j} ,\ 1 \leq i \leq k , $$

with vertex at the point $ ( x _ {0} , t _ {0} ) $ and a plane $ t = t _ {1} $. This subdomain is defined by the following property: small sufficiently smooth changes of the initial data inside $ D $ do not affect the value of the solution $ u $ at the point $ ( x _ {0} , t _ {0} ) $. In (1) it is assumed that $ L _ {ij} $ is a linear differential operator of order $ n _ {j} $ and that the order of the differentiations in it with respect to $ t $ does not exceed $ n _ {j} - 1 $. A "change inside" means a change in some domain that together with its boundary lies in $ D $.

For the wave equation

$$ \tag{2 } u _ {tt} - \sum _ { i= 1} ^ { n } u _ {x _ {i} x _ {i} } = 0 $$

the solution $ u $ of the Cauchy problem

$$ \tag{3 } \left . u \right | _ {t= 0} = \phi _ {0} ,\ \left . \frac{\partial u }{\partial t } \right | _ {t= 0} = \phi _ {1} $$

at the point $ ( x _ {0} , t _ {0} ) $, $ t _ {0} > 0 $, is completely determined by the values of the functions $ \phi _ {0} $ and $ \phi _ {1} $ on the sphere $ | y - x _ {0} | = t _ {0} $ for odd $ n > 1 $ and in the ball $ | y - x _ {0} | \leq t _ {0} $ for even $ n $ and $ n = 1 $, hence the domain $ | y - x _ {0} | < t _ {0} $ in the plane $ t = 0 $ is a lacuna for equation (2) for odd $ n > 1 $. For even $ n $ and for $ n = 1 $ 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 $ x _ {0} $ leads to a spherical wave with centre at this point, which for odd $ n > 1 $ has outward and inward facing fronts. For the remaining values of $ n $ 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 $ n $ of space variables is even (see [1]). The analogous question for $ n = 3 $ 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)

Comments

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 [3] 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].

References

[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