Minkowski problems arise from sub-linear elliptic equations

Let $\Omega \subset R^N$ be a bounded convex domain with boundary ∂Ω and $\nu \left(x\right)$ be the unit outer vector normal to ∂Ω at x. Let $S^{N-1}$ be the unit sphere in $R^N$. Then, the Gauss mapping $g:\partial \Omega \to S^{N-1}$, defined almost everywhere with respect to surface measure σ, is given by $g\left(x\right)=\nu \left(x\right)$. For $0<\beta <1$, it is well known that the following problem of sub-linear elliptic equation$\{\begin{array}{cc}-\Delta \phi ={\phi }^{\beta }& x\in \Omega \\ \phi >0& x\in \Omega \\ \phi =0& x\in \partial \Omega \end{array}$ has a unique solution. Moreover, it is easy to prove that each component of $\nabla \phi \left(x\right)$ is well-defined almost everywhere on ∂Ω with respect to σ. Therefore, we can assign a measure ${\mu }_{\Omega }$ on $S^{N-1}$ such that $d{\mu }_{\Omega }=g_{⁎}\left(\right|\nabla \phi {|}^{2}d\sigma )$. That is$\underset{S^{N-1}}{\int }f\left(\xi \right)d{\mu }_{\Omega }\left(\xi \right)=\underset{\partial \Omega }{\int }f\left(\nu \right(x\left)\right)\left|\nabla \phi \right(x){|}^{2}d\sigma$ for every $f\in C\left(S^{N-1}\right)$. The so-called Minkowski problem associated with ${\mu }_{\Omega }$ asks to find bounded convex domain Ω so that ${\mu }_{\Omega }=\mu$ for a given Borel measure μ on $S^{N-1}$. Our main results of this paper are the weak continuity of ${\mu }_{\Omega }$ with respect to Hausdorff metric and the unique solvability of Minkowski problem associated with ${\mu }_{\Omega }$. As a byproduct of our setting, an isoperimetric inequality is obtained.