Heat kernel

From Wikipedia, the free encyclopedia

In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a particular 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 most well-known heat kernel is the heat kernel of d-dimensional Euclidean space R^d, which has the form

$K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}.$

This solves the heat equation

$K_t(t,x,y) = \Delta_x K(t,x,y)\,$

for all t > 0 and x,y ∈ R^d, with the initial condition

$\lim_{t\downarrow 0} K(t,x,y) = \delta(x-y)=\delta_x(y)$

where δ is a Dirac delta distribution and the limit is taken in the sense of distributions. To wit, for every smooth function φ of compact support,

$\lim_{t\downarrow 0}\int_{\mathbf{R}^n} K(t,x,y)\phi(y)\,dy = \phi(x).$

On a more general domain Ω in R^d, 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

$K_t(t,x,y) = \Delta K(t,x,y) \rm{\ \ for\ all\ } t>0 \rm{\ and\ } x,y\in\Omega$

$\lim_{t\downarrow 0} K(t,x,y) = \delta_x(y)\rm{\ \ for\ all\ } x,y\in\Omega$

$K(t,x,y) = 0, \quad x\in\partial\Omega \rm{\ or\ } y\in\partial\Omega.$

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_n be the eigenvalues for the Dirichlet problem of the Laplacian

$\left\{ \begin{array}{ll} \Delta \phi + \lambda \phi = 0 & \mathrm{in\ }\ U\\ \phi=0 & \mathrm{on\ }\ \partial U. \end{array}\right.$

Let φ_n denote the associated eigenfunctions, normalized to be orthonormal in L²(U). The inverse Dirichlet Laplacian Δ^-1 is a compact and selfadjoint operator, and so the spectral theorem implies that the eigenvalues satisfy

$0 = \lambda_1 < \lambda_2\le \lambda_3\le\cdots,\quad \lambda_n\to\infty.$

The heat kernel has the following expression:

$K(t,x,y) = \sum_{n=0}^\infty e^{-\lambda_n t}\phi_n(x)\phi_n(y).$

(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, defined for compactly supported smooth φ by

$T\phi = \int_\Omega K(t,x,y)\phi(y)\,dy.$

The spectral mapping theorem gives a representation of T in the form

T = e t Δ .

[edit] References

Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag
Chavel, Isaac (1984), Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Boston, MA: Academic Press, MR 768584, ISBN 978-0-12-170640-1 .
Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9
Gilkey, Peter B. (1994), Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem, ISBN 978-0-8493-7874-4, http://www.emis.de/monographs/gilkey/

Heat kernel

From Wikipedia, the free encyclopedia

[edit] References

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