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Let $\Omega$ a bounded and open with Lipschitz boundary. I know that exists the trace operator in the case of this $\Omega$. My question is :

When $\Omega$ is bounded and open with Lipschitz boundary, the problem

$$ \left\{ \begin{array}{ccccccc} -\Delta u = 0 , \ in \ \Omega \\ \ u=g , \ in \ \partial\Omega \\ \end{array} \right. $$

have unique solution $u \in H^{1}(\Omega)$? (the equation is in the weak sense and the boundary condition is in the sense of the trace operator, that is , $g$ is in the image of the trace operator)

I dont found a book talking about of this problem in the case of Lipschitz boudary and nonzero boundary data.

Someone know a book with the answer ? (I believe the answer is yes because of the existence of the trace operator in this case) Thanks in advance!

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  • $\begingroup$ In order to get a unique solution you need to specify two things: 1. Unique in which space? $H^1_0(\Omega)$ or some space of smoother functions? 2. Where does $g$ belong? $H^{-1}(\partial\Omega)$? $\endgroup$ Feb 18 '14 at 19:23
  • $\begingroup$ You're right. I forget to write these details . sorry for i forget (my english is terrible , sorry) . i writed better the question $\endgroup$ Feb 18 '14 at 19:29
  • $\begingroup$ Have you tried to prove it? Hint: minimize the functional $I(u)=\int_\Omega |\nabla u|^2$ restricted to the set $\mathcal{K}=\{u\in H^1(\Omega):\ Tu=g\}$ where $T$ is the trace operator. I could not find it now, but I have answered this question somewhere. $\endgroup$
    – Tomás
    Feb 18 '14 at 19:38
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For $g\in H^{1/2}(\partial\Omega)$ you have uniqueness in $H^1(\Omega)$, provided that $\partial\Omega$ satisfies the cone condition - Lipschitz condition is even stronger, so with Lipschitz domains you also have uniqueness.

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    $\begingroup$ uhm -- the poster asked for a reference. $\endgroup$
    – Thomas
    Feb 18 '14 at 19:34
  • $\begingroup$ but every function of $L^2 (\partial \Omega)$ is in the image of the trace operator ? $\endgroup$ Feb 18 '14 at 19:36
  • $\begingroup$ No, the image of $T$ is $H^{1/2}(\partial\Omega)$, which is a proper subset of $L^2(\partial\Omega)$ $\endgroup$
    – Tomás
    Feb 18 '14 at 19:41

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