Heat equation in bounded domain

Question

Let $\Omega \subseteq \mathbb{R}^n$, $n\geq 2$, be a bounded domain with boundary $\partial \Omega \subseteq C^2,v$ outer unit normal vector on $\partial \Omega$, $h \in L^2(\Omega)$. Let $u \in C\left( \left[ 0,\infty \right);{{L}^{2}}\left( \Omega \right) \right)\cap C^1\left( \left( 0,\infty \right);{{H}^{2}}\left( \Omega \right) \right)$ be a solution to the Dirichlet problem $$ \left\{ \begin{matrix} {{u}_{t}}\left( t,x \right)={{\Delta }_{x}}u\left( t,x \right),t\in \left( 0,\infty \right),\ x\in \Omega \\ u\left( t,x \right)=0,t\in \left( 0,\infty \right),\ x\in \partial \Omega \\ u\left( 0,x \right)=h\left( x \right),\ x\in \Omega \\ \end{matrix} \right.$$

Let $w \in C\left( \left[ 0,\infty \right);{{L}^{2}}\left( \Omega \right) \right)\cap C^1\left( \left( 0,\infty \right);{{H}^{2}}\left( \Omega \right) \right)$ be a solution to the Neumann problem $$\left\{ \begin{matrix} {{w}_{t}}\left( t,x \right)={{\Delta }_{x}}w\left( t,x \right),t\in \left( 0,\infty \right),\ x\in \Omega \\ \frac{\partial w}{\partial v}\left( t,x \right)=0,t\in \left( 0,\infty \right),\ x\in \partial \Omega \\ w\left( 0,x \right)=h\left( x \right),\ x\in \Omega \\ \end{matrix} \right.$$ Show that there are bounded linear operators $ {{E}_{D}}\left( t \right):{{L}^{2}}\left( \Omega \right)\to {{L}^{2}}\left( \Omega \right)$ and $ {{E}_{N}}\left( t \right):{{L}^{2}}\left( \Omega \right)\to {{L}^{2}}\left( \Omega \right)$ such that $u\left( t,\centerdot \right)={{E}_{D}}\left( t \right)h$ and $w\left( t,\centerdot \right)={{E}_{N}}\left( t \right)h,t\in \left[ 0,\infty \right)$ and find their norms. Furthermore show that there are $0\ne {{h}_{D}}\in {{L}^{2}}\left( \Omega \right)$ y $0\ne {{h}_{N}}\in {{L}^{2}}\left( \Omega \right)$ such that ${{E}_{D}}\left( t \right)h_{D}=\left\| {{E}_{D}}\left( t \right) \right\|{{h}_{D}}$ y ${{E}_{N}}\left( t \right)h_{N}=\left\| {{E}_{N}}\left( t \right) \right\|{{h}_{N}}, \forall t\in \left[ 0,\infty \right)$. Describe those elements ${{h}_{D}}$ and ${{h}_{N}}$

Attempt: Let ${{\left\{ {{\lambda }_{n}^D} \right\}}_{n\in \mathbb{N}}}$ be eigenvalues with their respectives eigenfunctions ${{\left\{ {{\phi }_{n}^D} \right\}}_{n\in \mathbb{N}}}$ orthonormal basis of ${{L}^{2}}\left( \Omega \right)$, of the Dirichlet Laplacian. Therefore the solution is $u\left( t,x \right)=\sum\limits_{n\in \mathbb{N}}{{{\left\langle \phi _{n}^{D},h \right\rangle }_{{{L}^{2}}\left( \Omega \right)}}{{e}^{-\lambda _{n}^{D}t}}\phi _{n}^{D}\left( x \right)}$, so that ${{E}_{D}}\left( t \right)\left( f \right)=\sum\limits_{n\in \mathbb{N}}{{{\left\langle \phi _{n}^{D},f \right\rangle }_{{{L}^{2}}\left( \Omega \right)}}{{e}^{-\lambda _{n}^{D}t}}\phi _{n}^{D}\left( x \right)},f\in {{L}^{2}}\left( \Omega \right)$, but I don't know how to find the norm.

Answer 1

For the norm part, you have:

$$ \langle E_D(t) f, E_D(t)f\rangle = \sum_{n\in \mathbb{N}} e^{-2\lambda_n^Dt}\langle \phi_n^D,f\rangle^2 $$

by definition. So the norm, being

$$ \sup_{\|f\| = 1} \|E_D(t) f\| $$

is just given by

$$ e^{-\lambda_0^Dt} $$

where $\lambda_0^D$ is the smallest eigenvalue of the Dirichlet Laplacian. Furthermore, $h = \phi_0^D$ the corresponding eigenfunction would be the norm-achieving example desired.

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For the Neumann boundary condition case, note that $u \equiv c$ is a solution. It is clear that such a solution maximizes the norm for $E_N(t)$, that is given $h \equiv c$ you have $E_N(t)h = h = \|E_N(t)\|h$.