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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 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]].
 
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
+
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
  
<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>
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$$ \tag{1 }
  
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" />.
+
\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|wave equation]]
 
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>
+
$$ \tag{2 }
 +
u _ {tt} - \sum _ { i= 1} ^ { n }  u _ {x _ {i}  x _ {i} }  = 0
 +
$$
  
the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057060/l05706014.png" /> of the [[Cauchy problem|Cauchy problem]]
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the solution $  u $
 +
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>
+
$$ \tag{3 }
 +
\left . u \right | _ {t= 0}  = \phi _ {0} ,\  \left .  
 +
\frac{\partial  u }{\partial  t }
 +
\right | _ {t= 0}  = \phi _ {1}  $$
  
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.
+
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|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.
+
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 [[#References|[1]]]). The analogous question for $  n = 3 $
 +
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====
 
====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>
 
<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====
 
====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]]].
+
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====
 
====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>
 
<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>

Latest revision as of 14:30, 8 January 2022


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
How to Cite This Entry:
Lacuna. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lacuna&oldid=15246
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article