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A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

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Coercivity of $\int (\Delta u + u)^2$ on a subspace of $H^2$?

Let $\Omega = [0,L] \times [0,2\pi]$ and split its boundary into $\Gamma_d = \{0,L\} \times [0,2\pi]$, $\Gamma^1_p = [0,L] \times \{0\}$, $\Gamma^2_p = [0,L] \times\{2\pi\}$. Consider the following su …