In this paper, we compare the solutions of two Poisson PDE's in cylinders with Neumann boundary conditions, one with given initial data and one with data arranged decreasing in the $y-$direction. When the solutions are normalized to have zero mean, we show that the solution with symmetrized data is itself symmetrized and exhibits larger convex means. The main tools used are the $\star-$function introduced by Baernstein and a new subharmonicity result. As a consequence, we give a new proof of a conjecture of Kawohl for temperature gaps in rectangles.
Jeffrey J. Langford. "Subharmonicity, comparison results, and temperature gaps in cylindrical domains." Differential Integral Equations 29 (5/6) 493 - 512, May/June 2016. https://doi.org/10.57262/die/1457536888