3 problems

Let $(u,v)$ be an entire solution of the Keller–Segel system on $\mathbb{R}^2\times\mathbb{R}$, meaning that $u$ and $\nabla v$ are smooth and satisfy the system, with the second e…

Let $u$ be a solution of the Liouville equation … in $\mathbb{R}^2$. The concave-solution conjecture. If $u$ is concave, then $u$ is one-dimensional. The paper proves the analogous…

Let $u$ be a solution of the Liouville equation … in $\mathbb{R}^2$, and let $\delta$ be the standard Euclidean metric. Assume that $\mathbb{R}^2$ has diameter $pi$ under the metric…