View source for Lacuna

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

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

A lacuna in the theory of partial differential equations is a subdomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570601.png" /> of the intersection of the interior of the characteristic cone of a linear hyperbolic system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

with vertex at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570603.png" /> and a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570604.png" />. This subdomain is defined by the following property: small sufficiently smooth changes of the initial data inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570605.png" /> do not affect the value of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570606.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570607.png" />. In (1) it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570608.png" /> is a linear differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l0570609.png" /> and that the order of the differentiations in it with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706010.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706011.png" />. A  "change inside"  means a change in some domain that together with its boundary lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706012.png" />.

For the [[Wave equation|wave equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706014.png" /> of the [[Cauchy problem|Cauchy problem]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706017.png" />, is completely determined by the values of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706019.png" /> on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706020.png" /> for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706021.png" /> and in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706022.png" /> for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706024.png" />, hence the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706025.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706026.png" /> is a lacuna for equation (2) for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706027.png" />. For even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706028.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706029.png" /> equation (2) has no lacuna. This agrees with the [[Huygens principle|Huygens principle]] for solutions of the wave equation.

A perturbation of the initial data (3) in a small spherical neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706030.png" /> leads to a spherical wave with centre at this point, which for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706031.png" /> has outward and inward facing fronts. For the remaining values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706032.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706033.png" /> of space variables is even (see [[#References|[1]]]). The analogous question for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706034.png" /> was studied in [[#References|[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 [[#References|[3]]]). For the latter systems necessary and sufficient conditions of algebraic character have been obtained that ensure the presence of a lacuna.

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



====Comments====
Further research on lacunae for second-order equations was done by K.L. Stellmacher [[#References|[a1]]], R.G. Mclenaghan [[#References|[a2]]] and B. Ørsted [[#References|[a3]]]. Subsequent to the work [[#References|[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 [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.L. Stellmacher,   "Ein Beispiel einer Huyghensschen Differentialgleichung"  ''Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl.'' , '''10'''  (1953)  pp. 133–138</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Ørsted,   "The conformal invariance of Huygens' principle"  ''J. Diff. Geom.'' , '''16'''  (1981)  pp. 1–9</TD></TR><TR><TD valign="top">[a4a]</TD> <TD valign="top">  M.F. Atiyah,   R. Bott,   L. Gårding,   "Lacunas for hyperbolic differential operations with constant coefficients I"  ''Acta Math.'' , '''124'''  (1970)  pp. 109–189</TD></TR><TR><TD valign="top">[a4b]</TD> <TD valign="top">  M.F. Atiyah,   R. Bot,   L. Gårding,   "Lacunas for hyperbolic differential operations with constant coefficients II"  ''Acta Math.'' , '''131'''  (1973)  pp. 145–206</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Gårding,   "Sharp fronts of paired oscillatory integrals"  ''Publ. Res. Inst. Math. Sci. Kyoto Univ.'' , '''12. Suppl.'''  (1977)  pp. 53–68</TD></TR></table>