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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200501.png" /> be a compact [[Riemannian manifold|Riemannian manifold]] with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200502.png" />. Assume given a decomposition of the boundary as the disjoint union of two closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200504.png" />. Impose [[Neumann boundary conditions|Neumann boundary conditions]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200505.png" /> and [[Dirichlet boundary conditions|Dirichlet boundary conditions]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200506.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200507.png" /> be the temperature distribution of the manifold corresponding to an initial temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200508.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h1200509.png" /> is the solution to the equations:
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<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/h/h120/h120050/h12005010.png" /></td> </tr></table>
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Out of 47 formulas, 41 were replaced by TEX code.-->
  
<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/h/h120/h120050/h12005011.png" /></td> </tr></table>
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Let $M$ be a compact [[Riemannian manifold|Riemannian manifold]] with boundary $\partial M$. Assume given a decomposition of the boundary as the disjoint union of two closed sets $C _ { N }$ and $C _ { D }$. Impose [[Neumann boundary conditions|Neumann boundary conditions]] on $C _ { N }$ and [[Dirichlet boundary conditions|Dirichlet boundary conditions]] on $C _ { D }$. Let $u _ { \Phi }$ be the temperature distribution of the manifold corresponding to an initial temperature $\Phi$; $u _ { \Phi } ( x ; t )$ is the solution to the equations:
  
<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/h/h120/h120050/h12005012.png" /></td> </tr></table>
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\begin{equation*} ( \partial _ { t } + \Delta ) u = 0, \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005013.png" /> denotes differentiation with respect to the inward unit normal. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005014.png" /> be a smooth function giving the specific heat. The total heat energy content of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005015.png" /> is given by
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\begin{equation*} u ( x ; 0 ) = \Phi ( x ) , u _ { ; m } ( y ; t ) = 0 \text { for } y \in C _ { N } , t &gt; 0, \end{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/h/h120/h120050/h12005016.png" /></td> </tr></table>
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\begin{equation*} u ( y ; t ) = 0 \text { for } y \in C _ { D } , t &gt; 0. \end{equation*}
  
As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005017.png" />, there is an [[Asymptotic expansion|asymptotic expansion]]
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Here, $u_{:m}$ denotes differentiation with respect to the inward unit normal. Let $\rho$ be a smooth function giving the specific heat. The total heat energy content of $M$ is given by
  
<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/h/h120/h120050/h12005018.png" /></td> </tr></table>
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\begin{equation*} \beta ( \phi , \rho ) ( t ) = \int _ { M } u _ { \Phi } \rho. \end{equation*}
  
The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005019.png" /> are the heat content asymptotics and are locally computable.
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As $t \downarrow 0$, there is an [[Asymptotic expansion|asymptotic expansion]]
  
These coefficients were first studied with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005020.png" /> empty and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005021.png" />. Planar regions with smooth boundaries were studied in [[#References|[a5]]], [[#References|[a6]]], the upper hemisphere was studied in [[#References|[a4]]], [[#References|[a3]]], and polygonal domains in the plane were studied in [[#References|[a7]]]. See [[#References|[a11]]], [[#References|[a12]]] for recursive formulas on a general Riemannian manifold.
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\begin{equation*} \beta ( \phi , \rho ) ( t ) \sim \sum _ { n \geq 0 } \beta _ { n } ( \phi , \rho ) t ^ { n / 2 }. \end{equation*}
  
More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005022.png" /> be the [[Second fundamental form|second fundamental form]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005023.png" /> be the Riemann [[Curvature tensor|curvature tensor]]. Let indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005026.png" /> range from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005027.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005028.png" /> and index an orthonormal frame for the tangent bundle of the boundary. Let  ":"  (respectively, ";" ) denote covariant differentiation with respect to the [[Levi-Civita connection|Levi-Civita connection]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005029.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005030.png" />) summed over repeated indices. The first few coefficients have the form:
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The coefficients $\beta _ { n } ( \phi , \rho )$ are the heat content asymptotics and are locally computable.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005031.png" />;
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These coefficients were first studied with $C _ { N }$ empty and with $\phi = \rho = 1$. Planar regions with smooth boundaries were studied in [[#References|[a5]]], [[#References|[a6]]], the upper hemisphere was studied in [[#References|[a4]]], [[#References|[a3]]], and polygonal domains in the plane were studied in [[#References|[a7]]]. See [[#References|[a11]]], [[#References|[a12]]] for recursive formulas on a general Riemannian manifold.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005032.png" />;
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More generally, let $L$ be the [[Second fundamental form|second fundamental form]] and let $R$ be the Riemann [[Curvature tensor|curvature tensor]]. Let indices $a$, $b$, $c$ range from $1$ to $m - 1$ and index an orthonormal frame for the tangent bundle of the boundary. Let  ":" (respectively,  ";" ) denote covariant differentiation with respect to the [[Levi-Civita connection|Levi-Civita connection]] of $\partial M$ (respectively, of $M$) summed over repeated indices. The first few coefficients have the form:
  
<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/h/h120/h120050/h12005033.png" /></td> </tr></table>
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$\beta _ { 0 } ( \phi , \rho ) = \int _ { M } \phi \rho$;
  
<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/h/h120/h120050/h12005034.png" /></td> </tr></table>
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$\beta _ { 1 } ( \phi , \rho ) = - 2 \pi ^ { - 1 / 2 } \int _ { C _ { D } } \phi \rho$;
  
<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/h/h120/h120050/h12005035.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005033.png"/></td> </tr></table>
  
<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/h/h120/h120050/h12005036.png" /></td> </tr></table>
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\begin{equation*} + \int _ { C _ { N } } \phi _ { ; m } \rho \,d y; \end{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/h/h120/h120050/h12005037.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005035.png"/></td> </tr></table>
  
The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005038.png" /> is known.
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005036.png"/></td> </tr></table>
  
The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005040.png" /> have been determined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005041.png" /> is empty.
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\begin{equation*} + \frac { 4 } { 3 } \pi ^ { - 1 / 2 } \int _ { C _ { N } } \phi _ { ; m } \rho _ { ; m } d y. \end{equation*}
  
One can replace the [[Laplace operator|Laplace operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005042.png" /> by an arbitrary operator of Laplace type as the evolution operator [[#References|[a1]]], [[#References|[a2]]], [[#References|[a10]]], [[#References|[a9]]]. One can study non-minimal operators as the evolution operator, inhomogeneous boundary conditions, and time-dependent evolution operators of Laplace type. A survey of the field is given in [[#References|[a8]]].
+
The coefficient $\beta _ { 4 }$ is known.
 +
 
 +
The coefficients $\beta_5$ and $\beta_6$ have been determined if $C _ { D }$ is empty.
 +
 
 +
One can replace the [[Laplace operator|Laplace operator]] $\Delta$ by an arbitrary operator of Laplace type as the evolution operator [[#References|[a1]]], [[#References|[a2]]], [[#References|[a10]]], [[#References|[a9]]]. One can study non-minimal operators as the evolution operator, inhomogeneous boundary conditions, and time-dependent evolution operators of Laplace type. A survey of the field is given in [[#References|[a8]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. van den Berg,  S. Desjardins,  P. Gilkey,  "Functoriality and heat content asymptotics for operators of Laplace type"  ''Topol. Methods Nonlinear Anal.'' , '''2'''  (1993)  pp. 147–162</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. van den Berg,  P. Gilkey,  "Heat content asymptotics of a Riemannian manifold with boundary"  ''J. Funct. Anal.'' , '''120'''  (1994)  pp. 48–71</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. van den Berg,  P. Gilkey,  "Heat invariants for odd dimensional hemispheres"  ''Proc. R. Soc. Edinburgh'' , '''126A'''  (1996)  pp. 187–193</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. van den Berg,  "Heat equation on a hemisphere"  ''Proc. R. Soc. Edinburgh'' , '''118A'''  (1991)  pp. 5–12</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. van den Berg,  E.M. Davies,  "Heat flow out of regions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005043.png" />"  ''Math. Z.'' , '''202'''  (1989)  pp. 463–482</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. van den Berg,  J.-F. Le Gall,  "Mean curvature and the heat equation"  ''Math. Z.'' , '''215'''  (1994)  pp. 437–464</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. van den Berg,  S. Srisatkunarajah,  "Heat flow and Brownian motion for a region in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120050/h12005044.png" /> with a polygonal boundary"  ''Probab. Th. Rel. Fields'' , '''86'''  (1990)  pp. 41–52</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Gilkey,  "Heat content asymptotics"  Booss (ed.)  Wajciechowski (ed.) , ''Geometric Aspects of Partial Differential Equations'' , ''Contemp. Math.'' , '''242''' , Amer. Math. Soc.  (1999)  pp. 125–134</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D.M. McAvity,  "Surface energy from heat content asymptotics"  ''J. Phys. A: Math. Gen.'' , '''26'''  (1993)  pp. 823–830</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  D.M. McAvity,  "Heat kernel asymptotics for mixed boundary conditions"  ''Class. Quant. Grav'' , '''9'''  (1992)  pp. 1983–1998</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Savo,  "Uniform estimates and the whole asymptotic series of the heat content on manifolds"  ''Geom. Dedicata'' , '''73'''  (1998)  pp. 181–214</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A. Savo,  "Heat content and mean curvature"  ''J. Rend. Mat. Appl. VII Ser.'' , '''18'''  (1998)  pp. 197–219</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. van den Berg,  S. Desjardins,  P. Gilkey,  "Functoriality and heat content asymptotics for operators of Laplace type"  ''Topol. Methods Nonlinear Anal.'' , '''2'''  (1993)  pp. 147–162</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. van den Berg,  P. Gilkey,  "Heat content asymptotics of a Riemannian manifold with boundary"  ''J. Funct. Anal.'' , '''120'''  (1994)  pp. 48–71</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. van den Berg,  P. Gilkey,  "Heat invariants for odd dimensional hemispheres"  ''Proc. R. Soc. Edinburgh'' , '''126A'''  (1996)  pp. 187–193</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  M. van den Berg,  "Heat equation on a hemisphere"  ''Proc. R. Soc. Edinburgh'' , '''118A'''  (1991)  pp. 5–12</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. van den Berg,  E.M. Davies,  "Heat flow out of regions in ${\bf R} ^ { n }$"  ''Math. Z.'' , '''202'''  (1989)  pp. 463–482</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  M. van den Berg,  J.-F. Le Gall,  "Mean curvature and the heat equation"  ''Math. Z.'' , '''215'''  (1994)  pp. 437–464</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  M. van den Berg,  S. Srisatkunarajah,  "Heat flow and Brownian motion for a region in $\mathbf{R} ^ { 2 }$ with a polygonal boundary"  ''Probab. Th. Rel. Fields'' , '''86'''  (1990)  pp. 41–52</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  P. Gilkey,  "Heat content asymptotics"  Booss (ed.)  Wajciechowski (ed.) , ''Geometric Aspects of Partial Differential Equations'' , ''Contemp. Math.'' , '''242''' , Amer. Math. Soc.  (1999)  pp. 125–134</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  D.M. McAvity,  "Surface energy from heat content asymptotics"  ''J. Phys. A: Math. Gen.'' , '''26'''  (1993)  pp. 823–830</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  D.M. McAvity,  "Heat kernel asymptotics for mixed boundary conditions"  ''Class. Quant. Grav'' , '''9'''  (1992)  pp. 1983–1998</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  A. Savo,  "Uniform estimates and the whole asymptotic series of the heat content on manifolds"  ''Geom. Dedicata'' , '''73'''  (1998)  pp. 181–214</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  A. Savo,  "Heat content and mean curvature"  ''J. Rend. Mat. Appl. VII Ser.'' , '''18'''  (1998)  pp. 197–219</td></tr></table>

Revision as of 16:46, 1 July 2020

Let $M$ be a compact Riemannian manifold with boundary $\partial M$. Assume given a decomposition of the boundary as the disjoint union of two closed sets $C _ { N }$ and $C _ { D }$. Impose Neumann boundary conditions on $C _ { N }$ and Dirichlet boundary conditions on $C _ { D }$. Let $u _ { \Phi }$ be the temperature distribution of the manifold corresponding to an initial temperature $\Phi$; $u _ { \Phi } ( x ; t )$ is the solution to the equations:

\begin{equation*} ( \partial _ { t } + \Delta ) u = 0, \end{equation*}

\begin{equation*} u ( x ; 0 ) = \Phi ( x ) , u _ { ; m } ( y ; t ) = 0 \text { for } y \in C _ { N } , t > 0, \end{equation*}

\begin{equation*} u ( y ; t ) = 0 \text { for } y \in C _ { D } , t > 0. \end{equation*}

Here, $u_{:m}$ denotes differentiation with respect to the inward unit normal. Let $\rho$ be a smooth function giving the specific heat. The total heat energy content of $M$ is given by

\begin{equation*} \beta ( \phi , \rho ) ( t ) = \int _ { M } u _ { \Phi } \rho. \end{equation*}

As $t \downarrow 0$, there is an asymptotic expansion

\begin{equation*} \beta ( \phi , \rho ) ( t ) \sim \sum _ { n \geq 0 } \beta _ { n } ( \phi , \rho ) t ^ { n / 2 }. \end{equation*}

The coefficients $\beta _ { n } ( \phi , \rho )$ are the heat content asymptotics and are locally computable.

These coefficients were first studied with $C _ { N }$ empty and with $\phi = \rho = 1$. Planar regions with smooth boundaries were studied in [a5], [a6], the upper hemisphere was studied in [a4], [a3], and polygonal domains in the plane were studied in [a7]. See [a11], [a12] for recursive formulas on a general Riemannian manifold.

More generally, let $L$ be the second fundamental form and let $R$ be the Riemann curvature tensor. Let indices $a$, $b$, $c$ range from $1$ to $m - 1$ and index an orthonormal frame for the tangent bundle of the boundary. Let ":" (respectively, ";" ) denote covariant differentiation with respect to the Levi-Civita connection of $\partial M$ (respectively, of $M$) summed over repeated indices. The first few coefficients have the form:

$\beta _ { 0 } ( \phi , \rho ) = \int _ { M } \phi \rho$;

$\beta _ { 1 } ( \phi , \rho ) = - 2 \pi ^ { - 1 / 2 } \int _ { C _ { D } } \phi \rho$;

\begin{equation*} + \int _ { C _ { N } } \phi _ { ; m } \rho \,d y; \end{equation*}

\begin{equation*} + \frac { 4 } { 3 } \pi ^ { - 1 / 2 } \int _ { C _ { N } } \phi _ { ; m } \rho _ { ; m } d y. \end{equation*}

The coefficient $\beta _ { 4 }$ is known.

The coefficients $\beta_5$ and $\beta_6$ have been determined if $C _ { D }$ is empty.

One can replace the Laplace operator $\Delta$ by an arbitrary operator of Laplace type as the evolution operator [a1], [a2], [a10], [a9]. One can study non-minimal operators as the evolution operator, inhomogeneous boundary conditions, and time-dependent evolution operators of Laplace type. A survey of the field is given in [a8].

References

[a1] M. van den Berg, S. Desjardins, P. Gilkey, "Functoriality and heat content asymptotics for operators of Laplace type" Topol. Methods Nonlinear Anal. , 2 (1993) pp. 147–162
[a2] M. van den Berg, P. Gilkey, "Heat content asymptotics of a Riemannian manifold with boundary" J. Funct. Anal. , 120 (1994) pp. 48–71
[a3] M. van den Berg, P. Gilkey, "Heat invariants for odd dimensional hemispheres" Proc. R. Soc. Edinburgh , 126A (1996) pp. 187–193
[a4] M. van den Berg, "Heat equation on a hemisphere" Proc. R. Soc. Edinburgh , 118A (1991) pp. 5–12
[a5] M. van den Berg, E.M. Davies, "Heat flow out of regions in ${\bf R} ^ { n }$" Math. Z. , 202 (1989) pp. 463–482
[a6] M. van den Berg, J.-F. Le Gall, "Mean curvature and the heat equation" Math. Z. , 215 (1994) pp. 437–464
[a7] M. van den Berg, S. Srisatkunarajah, "Heat flow and Brownian motion for a region in $\mathbf{R} ^ { 2 }$ with a polygonal boundary" Probab. Th. Rel. Fields , 86 (1990) pp. 41–52
[a8] P. Gilkey, "Heat content asymptotics" Booss (ed.) Wajciechowski (ed.) , Geometric Aspects of Partial Differential Equations , Contemp. Math. , 242 , Amer. Math. Soc. (1999) pp. 125–134
[a9] D.M. McAvity, "Surface energy from heat content asymptotics" J. Phys. A: Math. Gen. , 26 (1993) pp. 823–830
[a10] D.M. McAvity, "Heat kernel asymptotics for mixed boundary conditions" Class. Quant. Grav , 9 (1992) pp. 1983–1998
[a11] A. Savo, "Uniform estimates and the whole asymptotic series of the heat content on manifolds" Geom. Dedicata , 73 (1998) pp. 181–214
[a12] A. Savo, "Heat content and mean curvature" J. Rend. Mat. Appl. VII Ser. , 18 (1998) pp. 197–219
How to Cite This Entry:
Heat content asymptotics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Heat_content_asymptotics&oldid=49991
This article was adapted from an original article by P.B. Gilkey (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article