# $W_{loc}^{1,2}$ functions on Alexandrov space approximated by functions of higher regularity?

M is an Alexandrov space with curvature bounded below.$\Omega$ is an open domain in M.Given a function $u \in W_{loc}^{1,2}\left( \Omega \right)$,we define a functional ${L_u}$ on $Li{p_0}\left( \Omega \right)$(Lipschitz functions with compact support) by $${L_u}\left( \phi \right) = - \int_\Omega {\left\langle {\nabla u,\nabla v} \right\rangle } dvol,\forall \phi \in Li{p_0}\left( \Omega \right)$$

Let $f \in {L^2}\left( \Omega \right)$ and $u \in W_{loc}^{1,2}\left( \Omega \right)$.If $${L_u}\left( \phi \right) \ge \int_\Omega {f\phi dvol} \left( {or{L_u}\left( \phi \right) \le \int_\Omega {f\phi dvol} } \right)$$ for all nonnegative $\phi \in Li{p_0}\left( \Omega \right)$,then according to Homander,the functional ${L_u}$ is a signed radon measure.In this case,u is said to be a subsolution (supersolution,resp.) of Possion equation ${L_u} = f \cdot vol$.It's solution if both sub and sup.

Assume $u \in W_{loc}^{1,2}\left( \Omega \right)$ and $f \in Lip\left( \Omega \right)$ satisfying ${L_u} = f \cdot vol$.Then by Zhang,Zhu's paper "Yau's gradient estimate on Alexandrov space". We have the regularity ${\left| {\nabla u} \right|^2} \in W_{loc}^{1,2}\left( \Omega \right)$ and $\left| {\nabla u} \right|$ is lower semi-continuous on $\Omega$.In this case,is there a sequence ${h_n}$,such that ${L_{{{\left| {\nabla {h_n}} \right|}^2}}}$ make sense and$\left| {\nabla {h_n}} \right| \to \left| {\nabla {{\left| {\nabla u} \right|}^2}} \right|$?

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