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Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain. Once every convex function is locally Lipschitz, we have that $\partial\Omega$ is Lipschitz, therefore, the trace operator $T: W^{1,2}(\Omega) \rightarrow L^2(\partial\Omega)$ is well defined.

Suppose that $\Omega \subset \{ x\in\mathbb{R}^N:\ x= (x_1,...,x_n),\ x_1 >0\}$ and $K \subset \partial \Omega$ is a compact set, with smooth boundary such that $K\subset \{ x\in\mathbb{R}^N : =(0,x_1,...,x_n)\}$.

Define $h(x) = 1$ if $x \in K$ and $h(x)=0$ if $ x \in \partial \Omega \setminus K$.

Is there a function $w \in W^{1,2}(\Omega)$ with $T(w) = h$? Intuitively this is true, but I don't know how to prove. Someone could help me to prove or disprove it? Thanks in advance!

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  • $\begingroup$ Do you know what is the image of the trace operator? $\endgroup$ – Tomás Aug 6 '14 at 23:45
  • $\begingroup$ i only know the definition $im T = T (W^{1,2}(\Omega))$ and is denoted by $H^{1\2} (\Omega)$. One time I heard that the image can be caracterized by Fourier transform. $\endgroup$ – math student Aug 7 '14 at 1:50
  • $\begingroup$ Why are you trying to prove this? $\endgroup$ – Tomás Aug 7 '14 at 12:05
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    $\begingroup$ I don't think that $h\in H^{1/2}(\partial\Omega)$, so if there is a solution for this problem, you will have to find another method to find it. You should study some book on Sobolev space, because your lack of knowledge in basic things will make things really hard for you. I suggest Leoni's book (see here amazon.com/Course-Sobolev-Graduate-Studies-Mathematics/dp/…) $\endgroup$ – Tomás Aug 7 '14 at 18:13
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    $\begingroup$ You can try to do some calculations by your own and see what happens. Here is one characterization of $H^{1/2}$ $$H^{1/2}(\partial\Omega)=\left\{f\in L^2({\partial \Omega}):\ \int_{\partial\Omega}\int_{\partial\Omega}\frac{|f(x)-f(y)|^2}{|x-y|^{N+1}}dxdy<\infty\right\}$$ $\endgroup$ – Tomás Aug 7 '14 at 18:15

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