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Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial \Omega$. Is there a continuous extension operator $P: L^2(\partial \Omega) \to H^1(\Omega), \quad Pg\vert_{\partial \Omega} = g \quad \text{ for } g \in L^2(\partial \Omega)$?

I thought about the weak solution operator of the harmonic Dirichlet problem, i.e. $$ \Delta u = 0 \text{ in } \Omega, \\ u = g \text{ on } \partial\Omega$$ and $u = Pg$, but I don't know if this operator is continuous.

DHtam
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  • Does this help, maybe? – Giuseppe Negro Mar 05 '19 at 13:39
  • The problem is that in these "inverse trace" theorems the trace space is $H^{1/2}$, but i need $L^2$. – DHtam Mar 05 '19 at 13:45
  • I suspect that $g\in L^2(\partial \Omega)$ is not enough regularity. You need $g\in H^{1/2}(\partial \Omega)$ for $g$ to be the trace of a $H^1(\Omega)$ function. – Giuseppe Negro Mar 05 '19 at 13:45
  • We wrote exactly the same thing. I am afraid that the solution map to the Dirichlet problem is not even well defined if $g$ is only $L^2(\partial \Omega)$. – Giuseppe Negro Mar 05 '19 at 13:46
  • See here, for example. Choosing $g\in L^2(\partial \Omega)$ you allow discontinuities, and the corresponding solution to the Dirichlet problem, if it exists, won't be in $H^1(\Omega)$. – Giuseppe Negro Mar 05 '19 at 14:28
  • Thank you. I don't want the $H^{1/2}(\partial \Omega)$ space, because the norm $\Vert u \Vert_{H^{1/2}(\partial \Omega)} = \inf_{v \in H^1(\Omega), v\vert_{\partial \Omega} = u} \Vert v \Vert_{H^1(\Omega)}$ gets unhandy in numerical computations. Do you know if I can equip the space $H^{1/2}(\partial \Omega)$ with a $L^2(\partial \Omega)$-inner product? – DHtam Mar 05 '19 at 14:42
  • This is another question, and a rather interesting one, I am also interested in numerically handling $H^{1/2}$. In a general domain, there's little hope, but if you have some symmetry there should be easier expressions for $|\cdot|_{H^{1/2}}$, based on fractional calculus. I recommend opening another thread with the new question. – Giuseppe Negro Mar 05 '19 at 14:56

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