On a more general domain Ω in Rd, 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 t > 0 on arbitrary domains and indeed on any Riemannian manifoldwith 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
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 λn be the eigenvalues for the Dirichlet problem of the Laplacian
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
There are several geometric results on heat kernels on manifolds; say, short-time asymptotics, long-time asymptotics, and upper/lower bounds of Gaussian type.