Heat semigroup representation

Question

It is known that the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega),$ (with $\Omega$ being a open subset of $R^n$) generates a $C_0-$semigroup in $L^2(\Omega)$). Moreover, in the case of the whole space $R^n$, the semigroup can be expressed by the means of the Gauss–Weierstrass kernel. I m wondering if this is also true for a bounded open subset $\Omega$ of $R^n$.

Answer 1

If you have a bounded region with a smooth boundary, and Dirichlet conditions, then $\Delta$ can be represented using an eigenfunction expansion, and the heat semigroup can be expressed in terms of the eigenfunctions of the Laplacian as well. That is, $$ -\Delta \varphi_n =\lambda_n \varphi_n, \;\;\; \|\varphi_n\|_{L^2(\Omega)}=1,\\ \lambda_1 \le \lambda_2 \le \lambda_3 \le \cdots, $$ and the eigenfunctions can be assumed to be mutually orthogonal. And the Heat semigroup $H(t)f=e^{t\Delta}f$, which is the solution of $\psi_{t}=\Delta \psi$ can be expressed in terms of the eigenfunctions of $-\Delta$, $$ H(t)f = \sum_{n=1}^{\infty}e^{-\sqrt{\lambda_n}t}\langle f,\varphi_n\rangle\varphi_n. $$ There would not generally be any simplification of this form. This is a $C^0$ semigroup on $L^2(\Omega)$.