# 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 $| y - x _ {|} < t _ {0}$ 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)