Subharmonicity, comparison results, and temperature gaps in cylindrical domains

PDE comparison principles for Robin problems

“…Much of the information gathered here also appears in Sections 2 and 3 of [31]. The major change in this subsection is that, here, we consider cylindrical domains rather than shell domains.…”

“…The techniques used in this paper are similar to those used by Baernstein in [8], the author in [29, 31] to prove Neumann versions of Theorems 1.1–1.3, and Baernstein, Laugesen, and the author in Chapter 10 of [10]. Indeed, the techniques employed here are drastically different from those used by Talenti to prove his original comparison result (and also those used in [3]).…”

“…See also [30] for a number of related comparison results on spheres and two-dimensional consequences in the plane via projection mappings. In [31], the author compares u and v when is a generalized cylinder and denotes the monotone decreasing rearrangement in the final variable (a one-dimensional Steiner symmetrization in one direction). Again, the author shows that the solutions compare via their convex means as in (1.2).…”

“…The structure of this subsection mimics the structure of the last one, because our goal is to gather all of the machinery necessary to prove our third main result, Theorem 1.3. Much of the information gathered here also appears in Sections 2 and 3 of [31]. The major change in this subsection is that, here, we consider cylindrical domains rather than shell domains.…”

“…We are now prepared to state our commutativity result for the cylindrical setting. See Proposition 2 of [31].…”