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I can't understand why $\|g\|_{H_{\partial}^{1/2}} = \|u\|_{H^1} $ where $u$ solves the problem \begin{equation*} - \Delta u + u = 0, \quad \Omega\\ u|_{\partial \Omega} = g \end{equation*}

The weak form should be to find $w \in H^1_0$:

\begin{equation*} (w,v)_{H^1} = -(G, v)_{H^1} , \quad \forall v \in H^1_0 \end{equation*}

where $ u = w+G$ and $G$ is the lifting of $g$.

$(.,.)_{H^1}$ is the $H^1$ inner-product.

Now I can only get this estimate from the Lax-Milgram lemma:

$$ \|u\| -\|G\| \leq \|w\| \leq \|G\| $$

And so:

$$ \|g\| \leq \|u\| \leq 2\|g\| $$

As the $H^{1/2}$ norm is the $\inf$

But how to get equality?

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Recall that $H^{1/2}(\Gamma):=\gamma_0(H^1(\Omega))$, where $\gamma_0(u):=u|_{\partial \Omega } $. Then we have that $\ker \gamma = H_0^1(\Omega)$. By the orthogonal descomposition $H^1(\Omega) = H^1_0(\Omega) \oplus H_0^1(\Omega)^{\perp}$. The restricted trace operator $\tilde {\gamma}_0:H_0^1(\Omega)^{\perp} \to H^{1/2}(\Gamma)$ is a linear isomorphism.

Now for any $g \in H^{1/2}(\Gamma)$ there is a $u = u_0+u^{\perp} \in H^1_0(\Omega) \oplus H_0^1(\Omega)^{\perp}$ with $g = \gamma_0(u)$. In particular $g=\gamma_0(u^{\perp})$, immediately we obtain $\tilde{\gamma}_0^{-1}(g)=u^{\perp}$, then

$$\|g\|_{1/2} = \inf_{u \in H^1(\Omega):\gamma_0(u)=g} \|u\|_{H^1}= \inf_{u \in H^1(\Omega):\gamma_0(u)=g} \|u_0+\tilde{\gamma}^{-1}(g)\|_{H^1}.$$ This infimum reachs when $u_0=0$, then $$\|g\|_{1/2} = \|\tilde{\gamma}^{-1}(g)\|_{H^1}=\|u^{\perp}\|_{H^1}$$

Finally, is not too complicate to show that $u^{\perp}$ is the weak solution, because \begin{align} H_0^1(\Omega)^{\perp} & = \{u^{\perp}:\int_{\Omega}(u^{\perp}w+\nabla u^{\perp}\cdot \nabla w ) = 0 \quad \forall w \in H^1_0(\Omega) \} \\ &= \{u^{\perp} \in \mathcal{D}'(\Omega):-\Delta u^{\perp}+u=0\}. \end{align}

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  • $\begingroup$ Thanks, do you recommend any books to learn about this kind of stuff, preferably with proofs? $\endgroup$
    – lucmobz
    Feb 23 at 6:48
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    $\begingroup$ Functional Analysis, Sobolev Spaces and Partial Differential Equations is an standard book, but I'm using a book in spanish about Sobolev Spaces. However the autor has a very similar book in english, with an stronger use of Sobolev spaces at work called Gatica - A Simple Introduction to the Mixed Finite Element Method Additionally take a look to Mclean - Strongly Elliptic Systems and Boundary Integral Equations. $\endgroup$ Feb 23 at 7:07

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