The concavity region of solutions to the Poisson equation in the plane
Jaime Arango, A. Gómez, J. Jiménez · Complex Variables and Elliptic Equations · 2019
We investigate the concavity region of solutions to the Poisson equation −Δu=1 on planar-bounded domains, by characterizing the nodal set D of the Hessian determinant of the solutions. In the case of solutions to the homogeneous Dirichlet problem on convex polygons, we prove that D is a smooth curve that touches every side of the polygon in exactly one point.