# Heat semigroup representation

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 $$R^n$$.

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 $$L^2(\Omega)$$.
• In fact I m interested with the continuity of the mapping $t\to \|S(t)x\|_{L^\infty(\Omega)},$ and I didn't succeed to do it using this formula. – Rabat Mar 17 at 16:54