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{{Short description|Fundamental solution to the heat equation, given boundary values}}
In the [[Mathematics|mathematical]] study of [[Physics:Heat conduction|heat conduction]] and [[Physics:Diffusion|diffusion]], a '''heat kernel''' is the [[Fundamental solution|fundamental solution]] to the [[Heat equation|heat equation]] on a specified domain with appropriate boundary conditions.  It is also one of the main tools in the study of the [[Spectral theory|spectrum]] of the [[Laplace operator]], and is thus of some auxiliary importance throughout [[Physics:Mathematical physics|mathematical physics]].  The heat kernel represents the evolution of [[Physics:Temperature|temperature]] in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time ''t''&nbsp;=&nbsp;0.

[[Image:Fundamental solution to the heat equation.gif|right|thumb|upright=2|Fundamental solution of the one-dimensional heat equation. Red: time course of <math>\Phi(x,t)</math>. Blue: time courses of <math>\Phi(x_0,t)</math> for two selected points. [https://www.geogebra.org/classic/SV6PruXx Interactive version.]]]
The most well-known heat kernel is the heat kernel of ''d''-dimensional [[Euclidean space]] '''R'''<sup>''d''</sup>, which has the form of a time-varying [[Gaussian function]],
:<math>K(t,x,y) = \exp\left(t\Delta\right)(x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-\|x-y\|^2/4t}\qquad(x,y\in\mathbb{R}^d,t>0)\,</math>
This solves the heat equation
:<math>\frac{\partial K}{\partial t}(t,x,y) = \Delta_x K(t,x,y)\,</math>
for all ''t''&nbsp;>&nbsp;0 and ''x'',''y''&nbsp;∈&nbsp;'''R'''<sup>''d''</sup>, where Δ is the Laplace operator, with the initial condition
:<math>\lim_{t \to 0} K(t,x,y) = \delta(x-y)=\delta_x(y)</math>
where δ is a Dirac delta distribution and the limit is taken in the sense of [[Distribution (mathematics)|distributions]].  To wit, for every smooth function ϕ of compact support,
:<math>\lim_{t \to 0}\int_{\mathbf{R}^d} K(t,x,y)\phi(y)\,dy = \phi(x).</math>

On a more general domain Ω in '''R'''<sup>''d''</sup>, such an explicit formula is not generally possible.  The next simplest cases of a disc or square involve, respectively, Bessel functions and Jacobi theta functions.  Nevertheless, the heat kernel (for, say, the [[Dirichlet problem]]) still exists and is smooth for ''t'' > 0 on arbitrary domains and indeed on any [[Riemannian manifold]] with boundary, provided the boundary is sufficiently regular.  More precisely, in these more general domains, the heat kernel for the Dirichlet problem is the solution of the initial boundary value problem

: <math>
\begin{align}
& \frac{\partial K}{\partial t}(t,x,y) = \Delta K(t,x,y) \text{ for all } t>0 \text{ and } x,y\in\Omega \\[6pt]
& \lim_{t \to 0} K(t,x,y) = \delta_x(y) \text{ for all } x,y\in\Omega \\[6pt]
& K(t,x,y) = 0, \quad x\in\partial\Omega \text{ or } y\in\partial\Omega.
\end{align}
</math>

It is not difficult to derive a formal expression for the heat kernel on an arbitrary domain.  Consider the Dirichlet problem in a connected domain (or manifold with boundary) ''U''.  Let ''λ''<sub>''n''</sub> be the eigenvalues for the Dirichlet problem of the Laplacian
:<math>\left\{
\begin{array}{ll}
\Delta \phi + \lambda \phi = 0 & \text{in } U\\
\phi=0 & \text{on }\ \partial U.
\end{array}\right.
</math>
Let ϕ<sub>''n''</sub> denote the associated [[Eigenfunction|eigenfunction]]s, normalized to be orthonormal in [[Lp space|L<sup>2</sup>(''U'')]].  The inverse Dirichlet Laplacian Δ<sup>−1</sup> is a [[Compact operator|compact]] and selfadjoint operator, and so the [[Spectral theorem|spectral theorem]] implies that the eigenvalues satisfy
:<math>0 < \lambda_1 < \lambda_2\le \lambda_3\le\cdots,\quad \lambda_n\to\infty.</math>
The heat kernel has the following expression:
{{NumBlk|:|<math>K(t,x,y) = \sum_{n=0}^\infty e^{-\lambda_n t}\phi_n(x)\phi_n(y).</math>|{{EquationRef|1}}}}
Formally differentiating the series under the sign of the summation shows that this should satisfy the heat equation.  However, convergence and regularity of the series are quite delicate.

The heat kernel is also sometimes identified with the associated [[Integral transform|integral transform]], defined for compactly supported smooth ϕ by
:<math>T\phi = \int_\Omega K(t,x,y)\phi(y)\,dy.</math>
The spectral mapping theorem gives a representation of ''T'' in the form
:<math>T = e^{t\Delta}.</math>

There are several geometric results on heat kernels on manifolds; say, short-time asymptotics, long-time asymptotics, and upper/lower bounds of Gaussian type.

==See also==

*[[Heat kernel signature]]
*[[Minakshisundaram–Pleijel zeta function]]
*[[Mehler kernel]]
*{{slink|Weierstrass transform#Generalizations}}

==References==

* {{Citation | last1=Berline | first1=Nicole | last2=Getzler | first2=E. | last3=Vergne | first3=Michèle | title=Heat Kernels and Dirac Operators | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | year=2004}}
* {{Citation | last1=Chavel | first1=Isaac | title=Eigenvalues in Riemannian geometry | publisher=[[Company:Academic Press|Academic Press]] | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-170640-1 | mr=768584 | year=1984 | volume=115}}.
* {{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | publisher=American Mathematical Society | location=Providence, R.I. | isbn=978-0-8218-0772-9 | year=1998}}
* {{Citation | last1=Gilkey | first1=Peter B. | title=Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem | url=http://www.emis.de/monographs/gilkey/ | isbn=978-0-8493-7874-4 | year=1994}}
*{{Citation | last1=Grigor'yan | first1=Alexander | title=Heat kernel and analysis on manifolds | url=https://books.google.com/books?id=X7QQcVa2EWsC | publisher=American Mathematical Society | location=Providence, R.I. | series=AMS/IP Studies in Advanced Mathematics | isbn=978-0-8218-4935-4 | mr=2569498 | year=2009 | volume=47}}

{{Functional analysis}}

{{DEFAULTSORT:Heat Kernel}}
[[Category:Heat conduction]]
[[Category:Spectral theory]]
[[Category:Parabolic partial differential equations]]

{{Sourceattribution|Heat kernel}}