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The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand : Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary $\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q): u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem $$ \int_Q A(x)\nabla u ...

Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$. Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...

Proposition 8.14. in Brezis states that:$(W_0^{1,p} (Ω))^*=W^{-1,p^*} (Ω)$ and we have the representation: $∀ F∈(W_0^{1,p} (Ω))^* ∃ f_0...f_n ∈L^{p^*} (Ω)$ such that $∀ u∈W_0^{1,p}(Ω)$ $F(u)=∫_Ω ...