Symmetry for an overdetermined boundary problem in a punctured domain

Monotonicity formulas in potential theory

“…Hence, the expansions (1.5) deprived of the reminder terms yield in this case explicit formulae for u, Du and DDu on the whole R n \ Ω. Tacking advantage of this observation, one easily realizes that the quantities are constant on rotationally symmetric solutions. Also notice en passant that the square of the first quantity is known in the literature as the P -function naturally associated with problem (1.1), see for instance [21,22,25,26,28,39,40]. In Subsection 1.3 we will shade some lights on the geometric nature of such a function.…”

Symmetry results for Serrin-type problems in doubly connected domains

“…Similar results also hold for quasi-linear possibly degenerate elliptic operators [5], fully nonlinear operators [32] and for the fractional laplacian [37]. For further symmetry results on this and related problems, see also [7,16,17,29,33].…”

A symmetry result for the p-Laplacian in a punctured manifold

“…When p = 2, the above problem on a Riemannian manifold appears in Physics as Electrostatics on anisotropic media [8], whereas for general p > 1 it arises in hydrodynamics, in the context of incompressible non-Newtonian fluids [11]. In Euclidean space, this problem has been recently studied [9] using a combination of the maximum principle and integral identities for P -functions which does not extend to curved manifolds.…”