Heat kernel

In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of 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 Template:Math.

Definition

Fundamental solution of the one-dimensional heat equation. Red: time course of $\mathrm{\Phi }(x,t)$. Blue: time courses of $\mathrm{\Phi }(x_0,t)$ for two selected points. Interactive version.

The most well-known heat kernel is the heat kernel of Template:Mvar-dimensional Euclidean space Template:Math, which has the form of a time-varying Gaussian function, $$K(t,x,y)=\frac{1}{{\left(4\pi t\right)}^{d/2}}\exp (-\frac{|x-y{|}^2}{4t}),$$ which is defined for all $x,y\in {ℝ}^d$ and $t>0$.Template:Sfn This solves the heat equation $$\{\begin{array}{cc}& \frac{\partial K}{\partial t}(t,x,y)={\mathrm{\Delta }}_{x}K(t,x,y)\\ & \underset{t\to 0}{\lim }K(t,x,y)=\delta (x-y)={\delta }_x(y)\end{array}$$ where Template:Math is a Dirac delta distribution and the limit is taken in the sense of distributions, that is, for every function Template:Math in the space Template:Math of smooth functions with compact support, we haveTemplate:Sfn $$\underset{t\to 0}{\lim }\underset{{ℝ}^d}{\int }K(t,x,y)\varphi (y)dy=\varphi (x).$$

On a more general domain Template:Math in Template:Math, 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 still exists and is smooth for Template:Math 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 the solution of the initial boundary value problem $$\{\begin{array}{cc}\frac{\partial K}{\partial t}(t,x,y)={\mathrm{\Delta }}_{x}K(t,x,y)& \text{for all }t>0\text{ and }x,y\in \mathrm{\Omega }\\ \underset{t\to 0}{\lim }K(t,x,y)={\delta }_x(y)& \text{for all }x,y\in \mathrm{\Omega }\\ K(t,x,y)=0& x\in \partial \mathrm{\Omega }\text{ or }y\in \partial \mathrm{\Omega }\end{array}$$

Spectral theory

Template:See also 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)Template:Math. Let Template:Math be the eigenvalues for the Dirichlet problem of the LaplacianTemplate:Sfn $$\{\begin{array}{cc}\mathrm{\Delta }\varphi +\lambda \varphi =0& \text{in }U,\\ \varphi =0& \text{on }\partial U.\end{array}$$ Let Template:Math denote the associated eigenfunctions, normalized to be orthonormal in [[Lp space|Template:Math]]. The inverse Dirichlet Laplacian Template:Math is a compact and selfadjoint operator, and so the spectral theorem implies that the eigenvalues of Template:Math satisfy $$0<{\lambda }_1\le {\lambda }_2\le {\lambda }_3\le \cdots ,{\lambda }_n\to \mathrm{\infty }.$$ The heat kernel has the following expression: $$K(t,x,y)=\sum _{n=0}^{\mathrm{\infty }}{e}^{-{\lambda }_{n}t}{\varphi }_n(x){\varphi }_n(y).$$ 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, defined for compactly supported smooth Template:Math by $$T\varphi =\underset{\mathrm{\Omega }}{\int }K(t,x,y)\varphi (y)dy.$$ The spectral mapping theorem gives a representation of Template:Math in the form the semigroupTemplate:SfnTemplate:Sfn

$$T=e^{t\mathrm{\Delta }}.$$

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
• Template:Slink

References

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