A Liouville type theorem for a variational problem with free boundary in three dimensions

We consider the minimum problem for the functional

_EΩ(u)=_∫Ω(|Du|²+λ²_χ{u>0})$E_{\Omega }\left(u\right)={\int }_{\Omega }({\left|Du\right|}^2+{\lambda }^2{\chi }_{\{u>0\}})$

in three dimensional space, where λ>0$\lambda >0$ is a constant. We will establish a Liouville type theorem for this variational problem: if u∈C(R³)$u\in C\left({\mathbb{R}}^3\right)$ is a nonnegative and nonzero global minimizer, then u(x)=λ((x−_x0)⋅ν)⁺$u\left(x\right)=\lambda {((x-x_0)\cdot \nu )}^{+}$ for some point _x0$x_0$ and some unit vector ν$\nu$.