View source for Heat content asymptotics

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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:

\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 &gt; 0, \end{equation*}

\begin{equation*} u ( y ; t ) = 0 \text { for } y \in C _ { D } , t &gt; 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|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 [[#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.

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:

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

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

<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>

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

<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>

<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>

\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|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====
<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>