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YCor
YCor
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Shifting SobolelvSobolev norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((0,T) \times \Omega)})$$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((0,T) \times \Omega)})$$ for all $u \in H^1((0,T) \times\Omega) \cap L^2((0,T);H^1_0(\Omega))$ and $\Box u \in L^2((0,T) \times \Omega)$. Can

Can one automatically deduce that the following estimate holds? $\|u\|_{L^2((0,T) \times\Omega)} \leq C (\|u\|_{L^2((0,T) \times \omega)} + \|\Box u\|_{H^{-1}((0,T) \times \Omega)})$$$\|u\|_{L^2((0,T) \times\Omega)} \leq C (\|u\|_{L^2((0,T) \times \omega)} + \|\Box u\|_{H^{-1}((0,T) \times \Omega)})$$

Shifting Sobolelv norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((0,T) \times \Omega)})$ for all $u \in H^1((0,T) \times\Omega) \cap L^2((0,T);H^1_0(\Omega))$ and $\Box u \in L^2((0,T) \times \Omega)$. Can one automatically deduce that the following estimate holds? $\|u\|_{L^2((0,T) \times\Omega)} \leq C (\|u\|_{L^2((0,T) \times \omega)} + \|\Box u\|_{H^{-1}((0,T) \times \Omega)})$

Shifting Sobolev norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((0,T) \times \Omega)})$$ for all $u \in H^1((0,T) \times\Omega) \cap L^2((0,T);H^1_0(\Omega))$ and $\Box u \in L^2((0,T) \times \Omega)$.

Can one automatically deduce that the following estimate holds? $$\|u\|_{L^2((0,T) \times\Omega)} \leq C (\|u\|_{L^2((0,T) \times \omega)} + \|\Box u\|_{H^{-1}((0,T) \times \Omega)})$$

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Ali
Ali
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Shifting Sobolelv norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate: $ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((0,T) \times \Omega)})$ for all $u \in H^1((0,T) \times\Omega) \cap L^2((0,T);H^1_0(\Omega))$ and $\Box u \in L^2((0,T) \times \Omega)$. Can one automatically deduce that the following estimate holds? $\|u\|_{L^2((0,T) \times\Omega)} \leq C (\|u\|_{L^2((0,T) \times \omega)} + \|\Box u\|_{H^{-1}((0,T) \times \Omega)})$