Let $u\in H^1_0((0,1)\times (0,T))$ due to the Poincaré inequality we have for all $t\in [0,T]$: $$||u(t)||_{L^2(0,1)} \leq ||u_x(t)||_{L^2(0,1)}$$ so in particular, for $t=0$ we will have: $$||u(0)||_{L^2(0,1)} \leq ||u_x(0)||_{L^2(0,1)}$$ is this statement true?
2026-03-25 15:49:16.1774453756
Poincaré inequlity
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