A toroidal tube solution to a problem involving mean curvature and Newtonian potential
This article deals with the following nonlocal geometric problem: find a set \(E\) in \(\mathbb R^2\) and a number \(\lambda\) such that the volume of \(E\) is \(m\) and the equation \(H(\partial E)+\gamma N(E)=\lambda\) is satisfied on the boundary of \(E\). Here \(m\), \(\gamma\) are given positive parameters, \(H(\partial E)\) is the mean curvature of \(\partial E\) and \(N(E)\) is the Newtonian potential operator \(N(E)(x)=\int_E\frac{dy}{4\pi |x-y|}\). The authors prove that for a sufficiently large \(\gamma\) there exists a solution of unit volume, which is closed to a set enclosed by a torus.
scientific article; zbMATH DE number 5903244