Heat kernel

 ( Heat Conduction ) 

In the mathematics 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 spectral theory 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 .

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Definition The most well-known heat kernel is the heat kernel of -dimensional Euclidean space , which has the form of a time-varying Gaussian function, $$K(t,x,y)\; =\; \backslash frac\{1\}\{\backslash left(4\backslash pi\; t\backslash right)^\{d/2\}\}\; \backslash exp\backslash left(-\backslash frac\{||x\; -\; y||^2\}\{4t\}\backslash right),$$ which is defined for all $x,y\backslash in\backslash mathbb\{R\}^d$ and $t\; >\; 0$. This solves the heat equation for the unknown function K. Here is a Dirac delta distribution, and the limit is taken in the sense of distributions, that is, for every function in the space of smooth functions with compact support, we have $$\backslash lim\_\{t\; \backslash to\; 0\}\backslash int\_\{\backslash mathbb\{R\}^d\}\; K(t,x,y)\backslash phi(y)\backslash ,dy\; =\; \backslash phi(x).$$

On a more general domain in , 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 function for 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

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Spectral theory 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). Let be the for the Dirichlet problem of the Laplacian Let denote the associated , normalized to be orthonormal in Lp space. The inverse Dirichlet Laplacian is a compact operator and selfadjoint operator, and so the spectral theorem implies that the eigenvalues of satisfy The heat kernel has the following expression: $$K(t,x,y)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; e^\{-\backslash lambda\_n\; t\}\backslash phi\_n(x)\backslash phi\_n(y).$$ 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 by $$T\backslash phi\; =\; \backslash int\_\backslash Omega\; K(t,x,y)\backslash phi(y)\backslash ,dy.$$ The spectral mapping theorem gives a representation of in the form the semigroup

$$T\; =\; e^\{t\backslash Delta\}.$$

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

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See also

• Heat kernel signature
• Minakshisundaram–Pleijel zeta function
• Mehler kernel

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Notes

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