Neumann comparison theorems in elliptic PDEs
In this paper, the author exemplary compares the solution of two PDEs, namely the Poisson PDE and the symmetrized Poisson PDE with Neumann boundary conditions with the aim to adopt Talenti's Theorem, since it cannot be applied directly. In order to do that, he starts with the background of the spherical and hemispherical setting. Next, he takes a look at the required rearrangements -- more precisely the \((k,n)\)-spherical and \((k,n)\)-hemispherical rearrangement to prove his Neumann comparison theorem for the Poisson PDE and further elliptic PDEs. After that, he also presents comparison results for the Euclidean space. In addition, the author states that ``This new technique of using projections to obtain original comparison results is one of the major contributions of our paper'' (see page 418). Finally, the paper is closed with some open problems and conjectures in the unit disk, where different examples concerning heat flow are discussed.
scientific article; zbMATH DE number 6536138
scientific article