Symmetry and monotonicity of positive solutions to elliptic equations with mixed boundary conditions in a kite

This paper investigates semilinear partial differential equations under Dirichlet-Neumann boundary conditions in a planar kite. By imposing specific geometric conditions on the kite, we establish that every positive solution exhibits symmetry with respect to the symmetry axis and maintains monotonicity along this axis. Our approach relies on the moving plane method, coupled with a strategic application of the maximum principle for mixed boundary conditions in a narrow domain.