Is the functional $F:H^{1}_{0}(\Omega) \longrightarrow \mathbb{R}$ given by \begin{equation} F(u) = \int_{\Omega} \langle (A_1(x)\chi_{\{u>0\}}+A_2(x)\chi_{\{u\le0\}}) \nabla u, \nabla u \rangle \, dx \end{equation} where $A_i,i=1,2$ is a matrix satisfying \begin{equation} \lambda |\xi|^2 \le \langle A_i(x) \xi,\xi \rangle \le \Lambda |\xi|^2, i=1,2. \end{equation} with $\lambda>0$ weak lower semicontinuous? I will appreciate any hint. Thank you.
Is the functional $F:H^{1}_{0}(\Omega) \longrightarrow \mathbb{R}$ given below weak lower semicontinuous?
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functional-analysis
sobolev-spaces
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0@LVK: The fate of my "counterexample" is getting darker now. – 2012-08-24