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Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\|_{L^2}^2,\\ \|(\varphi,\psi,w)\|_2^2&=\|\varphi_x\|_{L^2}^2+\|w_x\|_{L^2}^2+\|\psi_x\|_{L^2}^2. \end{align*} where $H_*^1(0,\ell)=\{f\in H^1;\;\int_0^\ell f(x)\;dx=0\}$ and $A$, $B$, $C$, $l$ are positive constants. It's also assumed $l\neq \frac{n\pi}{\ell}$, otherwise the condition $\|U\|_1=0\Rightarrow U=0$ is not satisfied.

Some research papers that deals with systems of PDEs from semigroup point of view (example) states $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. I've tried to prove it without success, so I'd like help.

It's pretty clear (from Poicaré inequality) that $\|\cdot\|_1\leq C\|\cdot\|_2$ however the converse is not obvious for me. Of course, if we know that $H$ is complete with respect to $\|\cdot\|_1$, (from open mapping theorem) it's not necessary to prove the converse but I'm having troubles to prove the completeness too.

If this question is not appropriate for the site, let me know and I will delete it.

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\|_{L^2}^2,\\ \|(\varphi,\psi,w)\|_2^2&=\|\varphi_x\|_{L^2}^2+\|w_x\|_{L^2}^2+\|\psi_x\|_{L^2}^2. \end{align*} where $H_*^1(0,\ell)=\{f\in H^1;\;\int_0^\ell f(x)\;dx=0\}$ and $A$, $B$, $C$, $l$ are positive constants. It's also assumed $l\neq \frac{n\pi}{\ell}$, otherwise the condition $\|U\|_1=0\Rightarrow U=0$ is not satisfied.

Some research papers that deals with systems of PDEs from semigroup point of view (example) states $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. I've tried to prove it without success, so I'd like help.

It's pretty clear (from Poincaré inequality) that $\|\cdot\|_1\leq K\|\cdot\|_2$ however the converse is not obvious for me. Of course, if we know that $H$ is complete with respect to $\|\cdot\|_1$, (from open mapping theorem) it's not necessary to prove the converse but I'm having troubles to prove the completeness too.

If this question is not appropriate for the site, let me know and I will delete it.

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\|_{L^2}^2,\\ \|(\varphi,\psi,w)\|_2^2&=\|\varphi_x\|_{L^2}^2+\|w_x\|_{L^2}^2+\|\psi_x\|_{L^2}^2. \end{align*} where $H_*^1(0,\ell)=\{f\in H^1;\;\int_0^\ell f(x)\;dx=0\}$ and $A$, $B$, $C$, $l$ are positive constants.

Some research papers that deals with systems of PDEs from semigroup point of view (example) states $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. I've tried to prove it without success, so I'd like help.

It's pretty clear (from Poicaré inequality) that $\|\cdot\|_1\leq C\|\cdot\|_2$ however the converse is not obvious for me. Of course, if we know that $H$ is complete with respect to $\|\cdot\|_1$, (from open mapping theorem) it's not necessary to prove the converse but I'm having troubles to prove the completeness too.

If this question is not appropriate for the site, let me know and I will delete it.

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\|_{L^2}^2,\\ \|(\varphi,\psi,w)\|_2^2&=\|\varphi_x\|_{L^2}^2+\|w_x\|_{L^2}^2+\|\psi_x\|_{L^2}^2. \end{align*} where $H_*^1(0,\ell)=\{f\in H^1;\;\int_0^\ell f(x)\;dx=0\}$ and $A$, $B$, $C$, $l$ are positive constants. It's also assumed $l\neq \frac{n\pi}{\ell}$, otherwise the condition $\|U\|_1=0\Rightarrow U=0$ is not satisfied.

Some research papers that deals with systems of PDEs from semigroup point of view (example) states $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. I've tried to prove it without success, so I'd like help.

It's pretty clear (from Poicaré inequality) that $\|\cdot\|_1\leq C\|\cdot\|_2$ however the converse is not obvious for me. Of course, if we know that $H$ is complete with respect to $\|\cdot\|_1$, (from open mapping theorem) it's not necessary to prove the converse but I'm having troubles to prove the completeness too.

If this question is not appropriate for the site, let me know and I will delete it.

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=c_1\|\varphi_x+\psi+c_2w\|_{L^2}^2+c_3\|w_x-c_2\varphi\|_{L^2}^2+c_4\|\psi_x\|_{L^2}^2,\\ \|(\varphi,\psi,w)\|_2^2&=\|\varphi_x\|_{L^2}^2+\|w_x\|_{L^2}^2+\|\psi_x\|_{L^2}^2. \end{align*} where $H_*^1(0,\ell)=\{f\in H^1;\;\int_0^\ell f(x)\;dx=0\}$.

Some research papers that deals with systems of PDEs from semigroup point of view (example) states $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. I've tried to prove it without success, so I'd like help.

It's pretty clear (from Poicaré inequality) that $\|\cdot\|_1\leq C\|\cdot\|_2$ however the converse is not obvious for me. Of course, if we know that $H$ is complete with respect to $\|\cdot\|_1$, (from open mapping theorem) it's not necessary to prove the converse but I'm having troubles to prove the completeness too.

If this question is not appropriate for the site, let me know and I will delete it.

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms \begin{align*} \|(\varphi,\psi,w)\|_1^2&=A\|\varphi_x+\psi+lw\|_{L^2}^2+B\|w_x-l\varphi\|_{L^2}^2+C\|\psi_x\|_{L^2}^2,\\ \|(\varphi,\psi,w)\|_2^2&=\|\varphi_x\|_{L^2}^2+\|w_x\|_{L^2}^2+\|\psi_x\|_{L^2}^2. \end{align*} where $H_*^1(0,\ell)=\{f\in H^1;\;\int_0^\ell f(x)\;dx=0\}$ and $A$, $B$, $C$, $l$ are positive constants.

Some research papers that deals with systems of PDEs from semigroup point of view (example) states $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent. I've tried to prove it without success, so I'd like help.

It's pretty clear (from Poicaré inequality) that $\|\cdot\|_1\leq C\|\cdot\|_2$ however the converse is not obvious for me. Of course, if we know that $H$ is complete with respect to $\|\cdot\|_1$, (from open mapping theorem) it's not necessary to prove the converse but I'm having troubles to prove the completeness too.

If this question is not appropriate for the site, let me know and I will delete it.