Minkowski problem arising from sub-linear elliptic equations

TL;DR

This paper studies a Minkowski problem linked to sublinear elliptic equations, establishing measure continuity and unique solvability for convex domains, advancing understanding in geometric analysis and PDEs.

Contribution

It introduces a new measure associated with sublinear elliptic equations and proves its weak continuity and the unique solvability of the related Minkowski problem.

Findings

The measure is weakly continuous with respect to the Hausdorff metric.

The Minkowski problem for this measure has a unique solution.

The results connect sublinear elliptic equations with geometric measure theory.

Abstract

For any bounded convex domain \Omega in R^N, we assign a positive finite Borel measure associated with the solution to a su-blinear elliptic equation in \Omega. We prove that this measure is weakly continuous in the sense of measure with respect Hausdorff metric, and the Minkowski problem associated with this measure is uniquely solvable.